PCENTER/PCENTER_S_context.csv

task_name,problem_type,instruction,instance,solution,obj,instance_variant,solution_variant,context_index,input_format,input_index_base
PCENTER,PCENTER,"On a map of the neighborhood the job is to pick a set number of bike parking hubs from a bunch of candidate spots and make sure every intersection is linked to a single open hub — nothing omitted and no intersection assigned to more than one hub. The way success is judged is by the longest walk anyone has to take: for each intersection record its walk to its hub, take the maximum of those walks, and aim to minimize that maximum. The concrete site list and distance numbers come next.

There are 5 candidate intersections; exactly 1 hubs must be opened, and their identifiers are 1, 2, 3, 4, 5.
From 1 to 2: walking distance 153.
From 1 to 3: walking distance 199.
From 1 to 4: walking distance 108.
From 1 to 5: walking distance 116.
From 2 to 1: walking distance 153.
From 2 to 3: walking distance 46.
From 2 to 4: walking distance 165.
From 2 to 5: walking distance 37.
From 3 to 1: walking distance 199.
From 3 to 2: walking distance 46.
From 3 to 4: walking distance 211.
From 3 to 5: walking distance 83.
From 4 to 1: walking distance 108.
From 4 to 2: walking distance 165.
From 4 to 3: walking distance 211.
From 4 to 5: walking distance 128.
From 5 to 1: walking distance 116.
From 5 to 2: walking distance 37.
From 5 to 3: walking distance 83.
From 5 to 4: walking distance 128.
Below are the precomputed walking distances between intersections.

Also, when you send the chosen plan back, please stick to this relaxed JSON layout so it's easy to read and parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

Think of ""selected"" as the list of candidate spots you decide to open, and ""assignments"" as, for each intersection in the input, which open spot that intersection is linked to. This is just a sketch of the shape I need, not the real answer — fill it with the actual identifiers from the instance.

Please use the identifiers exactly as they appear in the instance input — no renaming and no new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 153, 199, 108, 116], [153, 0, 46, 165, 37], [199, 46, 0, 211, 83], [108, 165, 211, 0, 128], [116, 37, 83, 128, 0]], 'p': 1, 'objective': 128.0}","{'facilities': [4], 'assignments': [4, 4, 4, 4, 4]}",128.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 153, 3: 199, 4: 108, 5: 116}}, {'id': 2, 'distances': {1: 153, 2: 0, 3: 46, 4: 165, 5: 37}}, {'id': 3, 'distances': {1: 199, 2: 46, 3: 0, 4: 211, 5: 83}}, {'id': 4, 'distances': {1: 108, 2: 165, 3: 211, 4: 0, 5: 128}}, {'id': 5, 'distances': {1: 116, 2: 37, 3: 83, 4: 128, 5: 0}}], 'objective': 128.0}","{'selected': [5], 'assignments': [5, 5, 5, 5, 5]}",1,markdown_table,1
PCENTER,PCENTER,"Many people in the coordination office are working out which temporary vaccination sites to open from a list of candidates, and then they’ll link every neighborhood to one of the open sites so no neighborhood is missed or assigned twice. The measure of success is straightforward: the plan that leaves the shortest possible longest trip wins — for any plan, look at each neighborhood’s travel time to its assigned site, pick the biggest one, and try to reduce that. There’s a strict limit on how many sites can be opened, and every neighborhood must be assigned to exactly one of those. The exact candidate locations, the travel times between places, and the number of sites allowed are shown below.

There are 5 locations in total, exactly 1 sites may be opened, and the ordered list of location identifiers is 0, 1, 2, 3, 4.
Travel time from 0 to 1 is 17.
Travel time from 0 to 2 is 33.
Travel time from 0 to 3 is 23.
Travel time from 0 to 4 is 31.
Travel time from 1 to 0 is 17.
Travel time from 1 to 2 is 25.
Travel time from 1 to 3 is 10.
Travel time from 1 to 4 is 29.
Travel time from 2 to 0 is 33.
Travel time from 2 to 1 is 25.
Travel time from 2 to 3 is 27.
Travel time from 2 to 4 is 34.
Travel time from 3 to 0 is 23.
Travel time from 3 to 1 is 10.
Travel time from 3 to 2 is 27.
Travel time from 3 to 4 is 25.
Travel time from 4 to 0 is 31.
Travel time from 4 to 1 is 29.
Travel time from 4 to 2 is 34.
Travel time from 4 to 3 is 25.
Use these travel times to assign every neighborhood to an open site and to evaluate the maximum assigned travel time that the plan seeks to minimize.

Oh, and when you reply with a plan, a handy little JSON sketch is easiest to parse. Something like this:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_site>, <assigned_site>, ...]
  }
}

""selected"" is just the list of candidate sites you decide to open. ""assignments"" lists, in the same order as the neighborhoods were given in the instance, which open site each neighborhood is linked to. Think of it as a simple form: which sites are open, and for each neighborhood which site they go to. This is only the expected shape — not the actual answer.

Please make sure to use the identifiers exactly as they appear in the instance input — do not rename or invent labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 17, 33, 23, 31], [17, 0, 25, 10, 29], [33, 25, 0, 27, 34], [23, 10, 27, 0, 25], [31, 29, 34, 25, 0]], 'p': 1, 'objective': 27.0}","{'facilities': [3], 'assignments': [3, 3, 3, 3, 3]}",27.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 17, 2: 33, 3: 23, 4: 31}}, {'id': 1, 'distances': {0: 17, 1: 0, 2: 25, 3: 10, 4: 29}}, {'id': 2, 'distances': {0: 33, 1: 25, 2: 0, 3: 27, 4: 34}}, {'id': 3, 'distances': {0: 23, 1: 10, 2: 27, 3: 0, 4: 25}}, {'id': 4, 'distances': {0: 31, 1: 29, 2: 34, 3: 25, 4: 0}}], 'objective': 27.0}","{'selected': [3], 'assignments': [3, 3, 3, 3, 3]}",2,markdown_table,0
PCENTER,PCENTER,"Picture this: there’s a list of potential pickup locker locations and a chart of travel distances between every pair of points. The task is to switch on a specific number of those sites as actual lockers, then link every customer address to exactly one of the opened lockers — no address skipped and no multiple links. To judge any setup, look at each customer’s distance to their assigned locker and take the biggest of those distances; the aim is to make that biggest distance as small as possible so no one has to travel too far. The exact locations, counts, and distances are shown below.

There are 5 candidate locations; you must open 2 lockers, and the location identifiers are A, B, C, D, E.
From A to B the travel distance is 12.
From A to C the travel distance is 57.
From A to D the travel distance is 57.
From A to E the travel distance is 38.
From B to A the travel distance is 12.
From B to C the travel distance is 45.
From B to D the travel distance is 57.
From B to E the travel distance is 26.
From C to A the travel distance is 57.
From C to B the travel distance is 45.
From C to D the travel distance is 66.
From C to E the travel distance is 51.
From D to A the travel distance is 57.
From D to B the travel distance is 57.
From D to C the travel distance is 66.
From D to E the travel distance is 63.
From E to A the travel distance is 38.
From E to B the travel distance is 26.
From E to C the travel distance is 51.
From E to D the travel distance is 63.
Assign each address to exactly one opened locker and minimize the maximum assigned travel distance.

Also, when you give the actual result, please stick to this simple JSON layout so it's easy to parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_site>, <assigned_site>, ...]
  }
}

This is just a sketch of the shape I expect, not the final answer itself. In plain terms: ""selected"" is the list of locker sites you decide to open, and ""assignments"" lists, for each original location in the instance, which opened site that location is linked to. Keep it light — think of ""selected"" as the checkboxes you tick, and ""assignments"" as the fill-in-the-blank that says which checkbox each address uses.

Quick reminder: use the exact identifiers from the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 12, 57, 57, 38], [12, 0, 45, 57, 26], [57, 45, 0, 66, 51], [57, 57, 66, 0, 63], [38, 26, 51, 63, 0]], 'p': 2, 'objective': 45.0}","{'facilities': [1, 3], 'assignments': [1, 1, 1, 3, 1]}",45.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 2, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 12, 'C': 57, 'D': 57, 'E': 38}}, {'id': 'B', 'distances': {'A': 12, 'B': 0, 'C': 45, 'D': 57, 'E': 26}}, {'id': 'C', 'distances': {'A': 57, 'B': 45, 'C': 0, 'D': 66, 'E': 51}}, {'id': 'D', 'distances': {'A': 57, 'B': 57, 'C': 66, 'D': 0, 'E': 63}}, {'id': 'E', 'distances': {'A': 38, 'B': 26, 'C': 51, 'D': 63, 'E': 0}}], 'objective': 45.0}","{'selected': ['B', 'D'], 'assignments': ['B', 'B', 'B', 'D', 'B']}",3,nl,names
PCENTER,PCENTER,"I’m helping the school transport officer figure out where to put a fixed number of bus stops from a list of suggested spots, and then tie each household to exactly one of the stops that get opened. The idea is simple: pick that set number of stops and assign every home to one open stop so that, when you look at everyone’s walk to their stop and take the longest single walk, that longest walk is as short as possible. Nobody can be left out or sent to two stops, and only the chosen stops can be used. The concrete distances and candidate spots will be shown below.

There are 8 locations, I need to open 1 stops, and the location IDs are 0, 1, 2, 3, 4, 5, 6, 7.
Distance from 0 to 1 is 37.
Distance from 0 to 2 is 52.
Distance from 0 to 3 is 56.
Distance from 0 to 4 is 40.
Distance from 0 to 5 is 57.
Distance from 0 to 6 is 57.
Distance from 0 to 7 is 37.
Distance from 1 to 0 is 37.
Distance from 1 to 2 is 40.
Distance from 1 to 3 is 38.
Distance from 1 to 4 is 24.
Distance from 1 to 5 is 40.
Distance from 1 to 6 is 49.
Distance from 1 to 7 is 34.
Distance from 2 to 0 is 52.
Distance from 2 to 1 is 40.
Distance from 2 to 3 is 45.
Distance from 2 to 4 is 31.
Distance from 2 to 5 is 45.
Distance from 2 to 6 is 61.
Distance from 2 to 7 is 20.
Distance from 3 to 0 is 56.
Distance from 3 to 1 is 38.
Distance from 3 to 2 is 45.
Distance from 3 to 4 is 26.
Distance from 3 to 5 is 28.
Distance from 3 to 6 is 59.
Distance from 3 to 7 is 36.
Distance from 4 to 0 is 40.
Distance from 4 to 1 is 24.
Distance from 4 to 2 is 31.
Distance from 4 to 3 is 26.
Distance from 4 to 5 is 28.
Distance from 4 to 6 is 40.
Distance from 4 to 7 is 22.
Distance from 5 to 0 is 57.
Distance from 5 to 1 is 40.
Distance from 5 to 2 is 45.
Distance from 5 to 3 is 28.
Distance from 5 to 4 is 28.
Distance from 5 to 6 is 61.
Distance from 5 to 7 is 38.
Distance from 6 to 0 is 57.
Distance from 6 to 1 is 49.
Distance from 6 to 2 is 61.
Distance from 6 to 3 is 59.
Distance from 6 to 4 is 40.
Distance from 6 to 5 is 61.
Distance from 6 to 7 is 48.
Distance from 7 to 0 is 37.
Distance from 7 to 1 is 34.
Distance from 7 to 2 is 20.
Distance from 7 to 3 is 36.
Distance from 7 to 4 is 22.
Distance from 7 to 5 is 38.
Distance from 7 to 6 is 48.
I'll use these distances and candidate spots to choose which 1 stops to open and assign every home so the maximum walk is minimized.

Just so we're on the same page, please return your pick-and-assign answer in a small JSON snippet like this — a relaxed template you can fill in.

{
  ""solution"": {
    ""selected"": [<stop_to_open>, <stop_to_open>, ...],
    ""assignments"": [<chosen_open_stop>, <chosen_open_stop>, ...]
  }
}

""selected"" is where you list the stops you decide to open. ""assignments"" is a matching list that says, for each household (in the same order they appear in the instance), which open stop that household is tied to. Think of it as a simple form: one line naming the open stops, and one line matching each home to one of them.

This is just the expected shape — a sketch, not the final filled-in answer.

Please make sure you use the exact identifiers from the instance input — do not rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 37, 52, 56, 40, 57, 57, 37], [37, 0, 40, 38, 24, 40, 49, 34], [52, 40, 0, 45, 31, 45, 61, 20], [56, 38, 45, 0, 26, 28, 59, 36], [40, 24, 31, 26, 0, 28, 40, 22], [57, 40, 45, 28, 28, 0, 61, 38], [57, 49, 61, 59, 40, 61, 0, 48], [37, 34, 20, 36, 22, 38, 48, 0]], 'p': 1, 'objective': 40.0}","{'facilities': [4], 'assignments': [4, 4, 4, 4, 4, 4, 4, 4]}",40.0,"{'problem_type': 'PCENTER', 'num_nodes': 8, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 37, 2: 52, 3: 56, 4: 40, 5: 57, 6: 57, 7: 37}}, {'id': 1, 'distances': {0: 37, 1: 0, 2: 40, 3: 38, 4: 24, 5: 40, 6: 49, 7: 34}}, {'id': 2, 'distances': {0: 52, 1: 40, 2: 0, 3: 45, 4: 31, 5: 45, 6: 61, 7: 20}}, {'id': 3, 'distances': {0: 56, 1: 38, 2: 45, 3: 0, 4: 26, 5: 28, 6: 59, 7: 36}}, {'id': 4, 'distances': {0: 40, 1: 24, 2: 31, 3: 26, 4: 0, 5: 28, 6: 40, 7: 22}}, {'id': 5, 'distances': {0: 57, 1: 40, 2: 45, 3: 28, 4: 28, 5: 0, 6: 61, 7: 38}}, {'id': 6, 'distances': {0: 57, 1: 49, 2: 61, 3: 59, 4: 40, 5: 61, 6: 0, 7: 48}}, {'id': 7, 'distances': {0: 37, 1: 34, 2: 20, 3: 36, 4: 22, 5: 38, 6: 48, 7: 0}}], 'objective': 40.0}","{'selected': [4], 'assignments': [4, 4, 4, 4, 4, 4, 4, 4]}",4,nl,0
PCENTER,PCENTER,"Recently the town decided to open a limited number of water refill stations and the question became which candidate locations to pick. The travel times between all spots are known, and each block has to be assigned to exactly one of the open stations — no skipping blocks and no overlapping assignments. What they’re trying to do is shrink the longest walk anyone faces: for any given plan, find the block that ends up farthest from its assigned station and reduce that distance as much as possible. The detailed map and numbers follow below.

There are 7 candidate locations, they may open 1 stations, and the location IDs are A, B, C, D, E, F, G.
From location A to B the travel distance is 43.
From location A to C the travel distance is 57.
From location A to D the travel distance is 57.
From location A to E the travel distance is 30.
From location A to F the travel distance is 40.
From location A to G the travel distance is 37.
From location B to A the travel distance is 43.
From location B to C the travel distance is 44.
From location B to D the travel distance is 46.
From location B to E the travel distance is 28.
From location B to F the travel distance is 22.
From location B to G the travel distance is 52.
From location C to A the travel distance is 57.
From location C to B the travel distance is 44.
From location C to D the travel distance is 53.
From location C to E the travel distance is 50.
From location C to F the travel distance is 28.
From location C to G the travel distance is 60.
From location D to A the travel distance is 57.
From location D to B the travel distance is 46.
From location D to C the travel distance is 53.
From location D to E the travel distance is 35.
From location D to F the travel distance is 33.
From location D to G the travel distance is 59.
From location E to A the travel distance is 30.
From location E to B the travel distance is 28.
From location E to C the travel distance is 50.
From location E to D the travel distance is 35.
From location E to F the travel distance is 27.
From location E to G the travel distance is 41.
From location F to A the travel distance is 40.
From location F to B the travel distance is 22.
From location F to C the travel distance is 28.
From location F to D the travel distance is 33.
From location F to E the travel distance is 27.
From location F to G the travel distance is 32.
From location G to A the travel distance is 37.
From location G to B the travel distance is 52.
From location G to C the travel distance is 60.
From location G to D the travel distance is 59.
From location G to E the travel distance is 41.
From location G to F the travel distance is 32.
Below are the pairwise travel distances they will use to assign each block to an open station.

One more thing — when you send back the plan, it'd be great if you stick to a simple JSON layout like this so it's easy to parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

Think of ""selected"" as the list of candidate spots we decide to open (placeholders above), and ""assignments"" as, for each block in the instance, which open spot that block is assigned to. Super casual: it's just a small form saying which sites are opened and where each block goes. This JSON is only a sketch of the shape I need, not the final answer itself.

Please be sure to use the exact identifiers given in the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 43, 57, 57, 30, 40, 37], [43, 0, 44, 46, 28, 22, 52], [57, 44, 0, 53, 50, 28, 60], [57, 46, 53, 0, 35, 33, 59], [30, 28, 50, 35, 0, 27, 41], [40, 22, 28, 33, 27, 0, 32], [37, 52, 60, 59, 41, 32, 0]], 'p': 1, 'objective': 40.0}","{'facilities': [5], 'assignments': [5, 5, 5, 5, 5, 5, 5]}",40.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 43, 'C': 57, 'D': 57, 'E': 30, 'F': 40, 'G': 37}}, {'id': 'B', 'distances': {'A': 43, 'B': 0, 'C': 44, 'D': 46, 'E': 28, 'F': 22, 'G': 52}}, {'id': 'C', 'distances': {'A': 57, 'B': 44, 'C': 0, 'D': 53, 'E': 50, 'F': 28, 'G': 60}}, {'id': 'D', 'distances': {'A': 57, 'B': 46, 'C': 53, 'D': 0, 'E': 35, 'F': 33, 'G': 59}}, {'id': 'E', 'distances': {'A': 30, 'B': 28, 'C': 50, 'D': 35, 'E': 0, 'F': 27, 'G': 41}}, {'id': 'F', 'distances': {'A': 40, 'B': 22, 'C': 28, 'D': 33, 'E': 27, 'F': 0, 'G': 32}}, {'id': 'G', 'distances': {'A': 37, 'B': 52, 'C': 60, 'D': 59, 'E': 41, 'F': 32, 'G': 0}}], 'objective': 40.0}","{'selected': ['F'], 'assignments': ['F', 'F', 'F', 'F', 'F', 'F', 'F']}",5,markdown_table,names
PCENTER,PCENTER,"There’s a simple relief puzzle on the table: open a fixed number of aid hubs from available spots, then attach every damaged neighborhood to exactly one hub so nobody’s missed or doubled up. The important measure is the longest journey anyone has to make — compute each neighborhood’s distance to its hub, then take the maximum of those distances, and try to bring that maximum down as much as possible. The map and the specific distances appear below.

# num_locations=5
# num_hubs_to_open=1
# location_ids=A, B, C, D, E
source_location,target_location,travel_distance
A,B,209
A,C,112
A,D,168
A,E,98
B,A,209
B,C,195
B,D,245
B,E,161
C,A,112
C,B,195
C,D,56
C,E,136
D,A,168
D,B,245
D,C,56
D,E,192
E,A,98
E,B,161
E,C,136
E,D,192

Also, when you send your solution back, please stick to this simple JSON layout so I can read it easily:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

This just sketches the shape I need: ""selected"" is the list of spots you decide to open as hubs, and ""assignments"" lists, for each neighborhood in the same order as the instance, which opened spot it's attached to. Think of it like filling out a form rather than writing code — it's just which hubs and who goes where.

One more thing: use the exact identifiers given in the problem instance — don't rename them or invent new labels. For example: Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 209, 112, 168, 98], [209, 0, 195, 245, 161], [112, 195, 0, 56, 136], [168, 245, 56, 0, 192], [98, 161, 136, 192, 0]], 'p': 1, 'objective': 192.0}","{'facilities': [4], 'assignments': [4, 4, 4, 4, 4]}",192.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 209, 'C': 112, 'D': 168, 'E': 98}}, {'id': 'B', 'distances': {'A': 209, 'B': 0, 'C': 195, 'D': 245, 'E': 161}}, {'id': 'C', 'distances': {'A': 112, 'B': 195, 'C': 0, 'D': 56, 'E': 136}}, {'id': 'D', 'distances': {'A': 168, 'B': 245, 'C': 56, 'D': 0, 'E': 192}}, {'id': 'E', 'distances': {'A': 98, 'B': 161, 'C': 136, 'D': 192, 'E': 0}}], 'objective': 192.0}","{'selected': ['E'], 'assignments': ['E', 'E', 'E', 'E', 'E']}",6,csv,names
PCENTER,PCENTER,"Recently the retail planning team faced a clear brief: from a set of shortlisted sites, pick exactly N dark stores to open, and make sure each delivery area is assigned to one and only one of the open stores. To grade any candidate plan, check each area’s travel distance to its assigned store and then look at the worst (longest) distance — the team wants that worst distance to be as small as possible. The specific sites, delivery zones, and travel distances are provided below.

# num_candidate_sites=8
# num_stores_to_open=3
# location_ids=1, 2, 3, 4, 5, 6, 7, 8
origin_location_id,destination_location_id,courier_travel_distance
1,2,35
1,3,22
1,4,31
1,5,30
1,6,45
1,7,38
1,8,28
2,1,35
2,3,29
2,4,34
2,5,31
2,6,43
2,7,36
2,8,31
3,1,22
3,2,29
3,4,24
3,5,25
3,6,33
3,7,29
3,8,15
4,1,31
4,2,34
4,3,24
4,5,9
4,6,47
4,7,29
4,8,25
5,1,30
5,2,31
5,3,25
5,4,9
5,6,46
5,7,28
5,8,24
6,1,45
6,2,43
6,3,33
6,4,47
6,5,46
6,7,35
6,8,46
7,1,38
7,2,36
7,3,29
7,4,29
7,5,28
7,6,35
7,8,28
8,1,28
8,2,31
8,3,15
8,4,25
8,5,24
8,6,46
8,7,28

When you send back the plan, please follow this simple JSON layout — here's a sketch of the shape I expect:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_site>, <assigned_site>, ...]
  }
}

""selected"" is just the list of the sites you choose to open. ""assignments"" lists, in order, which open site each delivery area is assigned to. Think of it like filling out a short form: pick the stores to open, then for every area say which of those stores it will use.

This is only the expected shape, not the actual answer — fill it in with the real identifiers from the instance.

Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”. Use those exact IDs from the input — no renaming and no new labels.","{'distance_matrix': [[0, 35, 22, 31, 30, 45, 38, 28], [35, 0, 29, 34, 31, 43, 36, 31], [22, 29, 0, 24, 25, 33, 29, 15], [31, 34, 24, 0, 9, 47, 29, 25], [30, 31, 25, 9, 0, 46, 28, 24], [45, 43, 33, 47, 46, 0, 35, 46], [38, 36, 29, 29, 28, 35, 0, 28], [28, 31, 15, 25, 24, 46, 28, 0]], 'p': 3, 'objective': 28.0}","{'facilities': [1, 5, 7], 'assignments': [7, 1, 7, 7, 7, 5, 7, 7]}",28.0,"{'problem_type': 'PCENTER', 'num_nodes': 8, 'num_open': 3, 'sites': [{'id': 1, 'distances': {1: 0, 2: 35, 3: 22, 4: 31, 5: 30, 6: 45, 7: 38, 8: 28}}, {'id': 2, 'distances': {1: 35, 2: 0, 3: 29, 4: 34, 5: 31, 6: 43, 7: 36, 8: 31}}, {'id': 3, 'distances': {1: 22, 2: 29, 3: 0, 4: 24, 5: 25, 6: 33, 7: 29, 8: 15}}, {'id': 4, 'distances': {1: 31, 2: 34, 3: 24, 4: 0, 5: 9, 6: 47, 7: 29, 8: 25}}, {'id': 5, 'distances': {1: 30, 2: 31, 3: 25, 4: 9, 5: 0, 6: 46, 7: 28, 8: 24}}, {'id': 6, 'distances': {1: 45, 2: 43, 3: 33, 4: 47, 5: 46, 6: 0, 7: 35, 8: 46}}, {'id': 7, 'distances': {1: 38, 2: 36, 3: 29, 4: 29, 5: 28, 6: 35, 7: 0, 8: 28}}, {'id': 8, 'distances': {1: 28, 2: 31, 3: 15, 4: 25, 5: 24, 6: 46, 7: 28, 8: 0}}], 'objective': 28.0}","{'selected': [2, 6, 8], 'assignments': [8, 2, 8, 8, 8, 6, 8, 8]}",7,csv,1
PCENTER,PCENTER,"Imagine the park has a handful of approved pads for portable toilets and a fixed quota of units to place; the job is to pick which spots to use and then tie each trailhead to exactly one of the opened toilets. The way to tell if a plan works well is to look at all the trailhead-to-restroom walks, find the longest one, and arrange placements so that this longest walk is as small as possible. The concrete candidate spots and distance numbers come next below.

# total_locations=7
# units_to_place=1
# location_ids=0, 1, 2, 3, 4, 5, 6
source_location_id,target_location_id,walking_distance
0,1,53
0,2,37
0,3,46
0,4,48
0,5,41
0,6,35
1,0,53
1,2,40
1,3,47
1,4,38
1,5,41
1,6,32
2,0,37
2,1,40
2,3,41
2,4,34
2,5,39
2,6,24
3,0,46
3,1,47
3,2,41
3,4,30
3,5,26
3,6,33
4,0,48
4,1,38
4,2,34
4,3,30
4,5,32
4,6,26
5,0,41
5,1,41
5,2,39
5,3,26
5,4,32
5,6,29
6,0,35
6,1,32
6,2,24
6,3,33
6,4,26
6,5,29

If you like, send the answer in this simple JSON shape so it's easy to read and parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

""selected"" is just the list of pad locations you decide to open (use the exact IDs from the instance). ""assignments"" is a list that tells, for each trailhead in the same order as the instance, which opened site that trailhead is tied to. Think of it like filling out a short form: which sites are opened, and which opened site each trailhead uses.

This JSON is only a sketch of the shape I expect, not the actual solution—fill in the real IDs from the instance when you reply.

Gentle reminder: use the exact identifiers from the instance input — no renaming and no new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 53, 37, 46, 48, 41, 35], [53, 0, 40, 47, 38, 41, 32], [37, 40, 0, 41, 34, 39, 24], [46, 47, 41, 0, 30, 26, 33], [48, 38, 34, 30, 0, 32, 26], [41, 41, 39, 26, 32, 0, 29], [35, 32, 24, 33, 26, 29, 0]], 'p': 1, 'objective': 35.0}","{'facilities': [6], 'assignments': [6, 6, 6, 6, 6, 6, 6]}",35.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 53, 2: 37, 3: 46, 4: 48, 5: 41, 6: 35}}, {'id': 1, 'distances': {0: 53, 1: 0, 2: 40, 3: 47, 4: 38, 5: 41, 6: 32}}, {'id': 2, 'distances': {0: 37, 1: 40, 2: 0, 3: 41, 4: 34, 5: 39, 6: 24}}, {'id': 3, 'distances': {0: 46, 1: 47, 2: 41, 3: 0, 4: 30, 5: 26, 6: 33}}, {'id': 4, 'distances': {0: 48, 1: 38, 2: 34, 3: 30, 4: 0, 5: 32, 6: 26}}, {'id': 5, 'distances': {0: 41, 1: 41, 2: 39, 3: 26, 4: 32, 5: 0, 6: 29}}, {'id': 6, 'distances': {0: 35, 1: 32, 2: 24, 3: 33, 4: 26, 5: 29, 6: 0}}], 'objective': 35.0}","{'selected': [6], 'assignments': [6, 6, 6, 6, 6, 6, 6]}",8,csv,0
PCENTER,PCENTER,"There’s a local planner deciding which lamp posts will become public Wi‑Fi spots — they can only switch on a predetermined number — and then each square or plaza in town will be connected to one of those switched‑on posts. To judge a setup, look at every plaza’s distance to its hotspot, find the single plaza that’s farthest away, and make plans so that that farthest distance is as small as it can be. No plaza is left out and none are split between hotspots. The concrete poles, plazas and distances appear below.

# total_candidate_locations=5
# hotspots_to_open=2
# location_ids=1, 2, 3, 4, 5
from_location_id,to_location_id,travel_distance
1,2,52
1,3,41
1,4,34
1,5,36
2,1,52
2,3,42
2,4,49
2,5,50
3,1,41
3,2,42
3,4,32
3,5,33
4,1,34
4,2,49
4,3,32
4,5,23
5,1,36
5,2,50
5,3,33
5,4,23

I'll keep the real answer in a tiny, predictable JSON shape so it's easy to plug into whatever checks you have. Here’s the sketch I’ll use:

{
  ""solution"": {
    ""selected"": [""<post_to_open>"", ""<post_to_open>"", ...],
    ""assignments"": [""<assigned_post>"", ""<assigned_post>"", ...]
  }
}

""selected"" lists which lamp posts we turn on as Wi‑Fi spots; ""assignments"" lists, for every plaza in the same order as the instance, which opened post that plaza connects to. This is just the expected shape — not the final choices.

Please use the exact identifiers from the instance input when you fill this in; do not rename them or invent new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 52, 41, 34, 36], [52, 0, 42, 49, 50], [41, 42, 0, 32, 33], [34, 49, 32, 0, 23], [36, 50, 33, 23, 0]], 'p': 2, 'objective': 34.0}","{'facilities': [1, 3], 'assignments': [3, 1, 3, 3, 3]}",34.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 2, 'sites': [{'id': 1, 'distances': {1: 0, 2: 52, 3: 41, 4: 34, 5: 36}}, {'id': 2, 'distances': {1: 52, 2: 0, 3: 42, 4: 49, 5: 50}}, {'id': 3, 'distances': {1: 41, 2: 42, 3: 0, 4: 32, 5: 33}}, {'id': 4, 'distances': {1: 34, 2: 49, 3: 32, 4: 0, 5: 23}}, {'id': 5, 'distances': {1: 36, 2: 50, 3: 33, 4: 23, 5: 0}}], 'objective': 34.0}","{'selected': [2, 4], 'assignments': [4, 2, 4, 4, 4]}",9,csv,1
PCENTER,PCENTER,"Someone on the food bank team has to pick which community centers to open for meal distribution — but only a certain number can be used — and then match each apartment complex to one of the centers that are opened. Every complex must have exactly one assigned site, and plans are compared by the single longest trip any resident would have to make (you find that by checking each complex’s assigned distance and taking the largest). The plan that makes that longest trip the shortest is the one they want. Full details of the candidate centers, apartment complexes, and distances appear below.

# total_locations_count=7
# centers_to_open=1
# location_ids=1, 2, 3, 4, 5, 6, 7
source_location_id,target_location_id,travel_distance
1,2,187
1,3,202
1,4,154
1,5,143
1,6,212
1,7,202
2,1,187
2,3,157
2,4,101
2,5,127
2,6,164
2,7,149
3,1,202
3,2,157
3,4,60
3,5,129
3,6,52
3,7,92
4,1,154
4,2,101
4,3,60
4,5,158
4,6,90
4,7,66
5,1,143
5,2,127
5,3,129
5,4,158
5,6,89
5,7,129
6,1,212
6,2,164
6,3,52
6,4,90
6,5,89
6,7,40
7,1,202
7,2,149
7,3,92
7,4,66
7,5,129
7,6,40

Also, just to keep things tidy, please follow this simple JSON layout when you send the plan back — nothing fancy, just this shape:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_site>, <assigned_site>, ...]
  }
}

This is just a sketch of the shape I need: ""selected"" is the list of community centers you choose to open, and ""assignments"" is, in the same order as the apartment complexes appear in the instance, which opened center each complex will use. Super casual: think of ""selected"" as the checkboxes you tick and ""assignments"" as the dropdown choices next to each complex.

Please make sure to use the exact identifiers from the instance input — do not rename them or invent new labels.  
Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'distance_matrix': [[0, 187, 202, 154, 143, 212, 202], [187, 0, 157, 101, 127, 164, 149], [202, 157, 0, 60, 129, 52, 92], [154, 101, 60, 0, 158, 90, 66], [143, 127, 129, 158, 0, 89, 129], [212, 164, 52, 90, 89, 0, 40], [202, 149, 92, 66, 129, 40, 0]], 'p': 1, 'objective': 158.0}","{'facilities': [4], 'assignments': [4, 4, 4, 4, 4, 4, 4]}",158.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 187, 3: 202, 4: 154, 5: 143, 6: 212, 7: 202}}, {'id': 2, 'distances': {1: 187, 2: 0, 3: 157, 4: 101, 5: 127, 6: 164, 7: 149}}, {'id': 3, 'distances': {1: 202, 2: 157, 3: 0, 4: 60, 5: 129, 6: 52, 7: 92}}, {'id': 4, 'distances': {1: 154, 2: 101, 3: 60, 4: 0, 5: 158, 6: 90, 7: 66}}, {'id': 5, 'distances': {1: 143, 2: 127, 3: 129, 4: 158, 5: 0, 6: 89, 7: 129}}, {'id': 6, 'distances': {1: 212, 2: 164, 3: 52, 4: 90, 5: 89, 6: 0, 7: 40}}, {'id': 7, 'distances': {1: 202, 2: 149, 3: 92, 4: 66, 5: 129, 6: 40, 7: 0}}], 'objective': 158.0}","{'selected': [5], 'assignments': [5, 5, 5, 5, 5, 5, 5]}",10,csv,1
PCENTER,PCENTER,"Many people are tired of long walks to recycle, so the task is to pick a specified number of lots to turn into drop-off points from a list of candidates and connect each block to one of those open points (one block, one point). To compare options, compute every block’s walk to its linked drop-off, take the worst (the farthest) of those walks, and prefer the setup where that worst walk is minimized — in other words, make the single longest walk as short as possible. The particulars — which lots, the distances, and how many to open — are given below.

{
  ""total_locations_count"": 6,
  ""num_dropoff_to_open"": 2,
  ""location_id_list"": [
    1,
    2,
    3,
    4,
    5,
    6
  ],
  ""data"": [
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 2,
      ""walk_distance_between_locations"": 28
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 3,
      ""walk_distance_between_locations"": 42
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 4,
      ""walk_distance_between_locations"": 38
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 5,
      ""walk_distance_between_locations"": 45
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 6,
      ""walk_distance_between_locations"": 43
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 1,
      ""walk_distance_between_locations"": 28
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 3,
      ""walk_distance_between_locations"": 47
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 4,
      ""walk_distance_between_locations"": 36
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 5,
      ""walk_distance_between_locations"": 35
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 6,
      ""walk_distance_between_locations"": 33
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 1,
      ""walk_distance_between_locations"": 42
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 2,
      ""walk_distance_between_locations"": 47
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 4,
      ""walk_distance_between_locations"": 35
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 5,
      ""walk_distance_between_locations"": 35
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 6,
      ""walk_distance_between_locations"": 39
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 1,
      ""walk_distance_between_locations"": 38
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 2,
      ""walk_distance_between_locations"": 36
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 3,
      ""walk_distance_between_locations"": 35
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 5,
      ""walk_distance_between_locations"": 53
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 6,
      ""walk_distance_between_locations"": 51
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 1,
      ""walk_distance_between_locations"": 45
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 2,
      ""walk_distance_between_locations"": 35
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 3,
      ""walk_distance_between_locations"": 35
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 4,
      ""walk_distance_between_locations"": 53
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 6,
      ""walk_distance_between_locations"": 42
    },
    {
      ""origin_location_id"": 6,
      ""destination_location_id"": 1,
      ""walk_distance_between_locations"": 43
    },
    {
      ""origin_location_id"": 6,
      ""destination_location_id"": 2,
      ""walk_distance_between_locations"": 33
    },
    {
      ""origin_location_id"": 6,
      ""destination_location_id"": 3,
      ""walk_distance_between_locations"": 39
    },
    {
      ""origin_location_id"": 6,
      ""destination_location_id"": 4,
      ""walk_distance_between_locations"": 51
    },
    {
      ""origin_location_id"": 6,
      ""destination_location_id"": 5,
      ""walk_distance_between_locations"": 42
    }
  ]
}

When you send your answer, just toss it into a tiny JSON blob like this so it's easy to read and parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_site>, <assigned_site>, ...]
  }
}

Keep in mind: ""selected"" is the list of lots you turn into drop-off points, and ""assignments"" lists, for each block in the same order as the input, which opened lot that block uses. Super simple — think of it like filling out a short form, not a report.

This JSON is only a sketch of the shape I expect, not the actual answer.

All identifiers in your final answer must match the instance input exactly — do not rename them or invent new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 28, 42, 38, 45, 43], [28, 0, 47, 36, 35, 33], [42, 47, 0, 35, 35, 39], [38, 36, 35, 0, 53, 51], [45, 35, 35, 53, 0, 42], [43, 33, 39, 51, 42, 0]], 'p': 2, 'objective': 35.0}","{'facilities': [1, 2], 'assignments': [1, 1, 2, 2, 2, 1]}",35.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 2, 'sites': [{'id': 1, 'distances': {1: 0, 2: 28, 3: 42, 4: 38, 5: 45, 6: 43}}, {'id': 2, 'distances': {1: 28, 2: 0, 3: 47, 4: 36, 5: 35, 6: 33}}, {'id': 3, 'distances': {1: 42, 2: 47, 3: 0, 4: 35, 5: 35, 6: 39}}, {'id': 4, 'distances': {1: 38, 2: 36, 3: 35, 4: 0, 5: 53, 6: 51}}, {'id': 5, 'distances': {1: 45, 2: 35, 3: 35, 4: 53, 5: 0, 6: 42}}, {'id': 6, 'distances': {1: 43, 2: 33, 3: 39, 4: 51, 5: 42, 6: 0}}], 'objective': 35.0}","{'selected': [2, 3], 'assignments': [2, 2, 3, 3, 3, 2]}",11,json,1
PCENTER,PCENTER,"We run last-mile deliveries and have a handful of candidate addresses to turn into micro hubs; we also know how far it is between any two places on the map. The task is to decide which predetermined number of sites to activate as hubs, and then assign every delivery zone to a single open hub (each zone goes to one hub, nothing duplicated). The runner-up metric is the worst courier trip — the single longest distance from a zone to its assigned hub — and the aim is to make that longest distance as small as we can. Concrete details follow below.

# total_candidate_addresses=6
# num_hubs_to_activate=1
# location_identifiers=0, 1, 2, 3, 4, 5
from_location_id,to_location_id,travel_distance
0,1,28
0,2,14
0,3,31
0,4,17
0,5,33
1,0,28
1,2,29
1,3,34
1,4,25
1,5,31
2,0,14
2,1,29
2,3,35
2,4,27
2,5,41
3,0,31
3,1,34
3,2,35
3,4,29
3,5,37
4,0,17
4,1,25
4,2,27
4,3,29
4,5,41
5,0,33
5,1,31
5,2,41
5,3,37
5,4,41

Just drop your answer into this simple JSON shape so it's easy to parse — list the sites you open and then, for every zone in the instance order, the open site that zone gets assigned to:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

""selected"" is the list of addresses we flip on as micro-hubs. ""assignments"" is a one-for-one list (same order as the instance's locations) saying which open site each location uses. Super casual: think of it like a form with two fields — chosen hubs, and then each zone's chosen hub.

This JSON is just the expected shape — a sketch, not the final filled-in plan.

Important: use the exact identifiers given in the problem input for any location labels — do not rename them or invent new ones. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 28, 14, 31, 17, 33], [28, 0, 29, 34, 25, 31], [14, 29, 0, 35, 27, 41], [31, 34, 35, 0, 29, 37], [17, 25, 27, 29, 0, 41], [33, 31, 41, 37, 41, 0]], 'p': 1, 'objective': 33.0}","{'facilities': [0], 'assignments': [0, 0, 0, 0, 0, 0]}",33.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 28, 2: 14, 3: 31, 4: 17, 5: 33}}, {'id': 1, 'distances': {0: 28, 1: 0, 2: 29, 3: 34, 4: 25, 5: 31}}, {'id': 2, 'distances': {0: 14, 1: 29, 2: 0, 3: 35, 4: 27, 5: 41}}, {'id': 3, 'distances': {0: 31, 1: 34, 2: 35, 3: 0, 4: 29, 5: 37}}, {'id': 4, 'distances': {0: 17, 1: 25, 2: 27, 3: 29, 4: 0, 5: 41}}, {'id': 5, 'distances': {0: 33, 1: 31, 2: 41, 3: 37, 4: 41, 5: 0}}], 'objective': 33.0}","{'selected': [0], 'assignments': [0, 0, 0, 0, 0, 0]}",12,csv,0
PCENTER,PCENTER,"Back in the planning meeting they gave a shortlist of possible bike dock spots and said only a certain number could be activated, so the task is to choose which ones to open and then link every neighborhood centroid to a single open dock. Every neighborhood must have exactly one assigned dock, and the performance metric is the biggest distance anyone has to travel — compute each neighborhood’s distance to its assigned dock, find the largest of those distances, and make that largest distance as small as possible. Specifics on locations and distances are shown below.

# num_candidate_dock_locations=7
# num_docks_to_open=1
# location_ids=1, 2, 3, 4, 5, 6, 7
from_location_id,to_location_id,travel_distance
1,2,43
1,3,50
1,4,44
1,5,27
1,6,49
1,7,43
2,1,43
2,3,51
2,4,32
2,5,46
2,6,47
2,7,55
3,1,50
3,2,51
3,4,50
3,5,45
3,6,42
3,7,42
4,1,44
4,2,32
4,3,50
4,5,36
4,6,57
4,7,44
5,1,27
5,2,46
5,3,45
5,4,36
5,6,42
5,7,54
6,1,49
6,2,47
6,3,42
6,4,57
6,5,42
6,7,62
7,1,43
7,2,55
7,3,42
7,4,44
7,5,54
7,6,62

Also, when you send back the plan, it's handy to use a simple JSON layout like this so it's easy to read and parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_dock>, <assigned_dock>, ...]
  }
}

Here ""selected"" is where you list which dock sites you decide to open (one placeholder per opened site), and ""assignments"" is a list that says which opened dock each neighborhood is linked to (in the same order as the neighborhoods in the instance). Think of it as a short form: the first array is the chosen docks, the second array assigns every neighborhood to one of those chosen docks.

This JSON is just a sketch of the shape I expect, not the actual solution. Please be sure to use the exact identifiers from the instance input — do not rename them or invent new labels. 

- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 43, 50, 44, 27, 49, 43], [43, 0, 51, 32, 46, 47, 55], [50, 51, 0, 50, 45, 42, 42], [44, 32, 50, 0, 36, 57, 44], [27, 46, 45, 36, 0, 42, 54], [49, 47, 42, 57, 42, 0, 62], [43, 55, 42, 44, 54, 62, 0]], 'p': 1, 'objective': 50.0}","{'facilities': [0], 'assignments': [0, 0, 0, 0, 0, 0, 0]}",50.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 43, 3: 50, 4: 44, 5: 27, 6: 49, 7: 43}}, {'id': 2, 'distances': {1: 43, 2: 0, 3: 51, 4: 32, 5: 46, 6: 47, 7: 55}}, {'id': 3, 'distances': {1: 50, 2: 51, 3: 0, 4: 50, 5: 45, 6: 42, 7: 42}}, {'id': 4, 'distances': {1: 44, 2: 32, 3: 50, 4: 0, 5: 36, 6: 57, 7: 44}}, {'id': 5, 'distances': {1: 27, 2: 46, 3: 45, 4: 36, 5: 0, 6: 42, 7: 54}}, {'id': 6, 'distances': {1: 49, 2: 47, 3: 42, 4: 57, 5: 42, 6: 0, 7: 62}}, {'id': 7, 'distances': {1: 43, 2: 55, 3: 42, 4: 44, 5: 54, 6: 62, 7: 0}}], 'objective': 50.0}","{'selected': [1], 'assignments': [1, 1, 1, 1, 1, 1, 1]}",13,csv,1
PCENTER,PCENTER,"Someone in charge of community health needs to pick a set number of mobile testing locations from a list of possible venues, then link every residential sector to one of the opened locations so nobody’s left out or double-assigned, and the priority is to cut down the single longest travel distance residents must make. To evaluate any choice, look up each sector’s distance to its assigned site and take the biggest value — the goal is to make that biggest value as small as possible. The detailed list of venues and distances is shown below.

There are 7 locations in this instance; exactly 1 sites must be opened, and the listed location identifiers are 1, 2, 3, 4, 5, 6, 7.
For the link with origin 1 and destination 2, the travel distance is 38.
For the link with origin 1 and destination 3, the travel distance is 47.
For the link with origin 1 and destination 4, the travel distance is 24.
For the link with origin 1 and destination 5, the travel distance is 56.
For the link with origin 1 and destination 6, the travel distance is 29.
For the link with origin 1 and destination 7, the travel distance is 42.
For the link with origin 2 and destination 1, the travel distance is 38.
For the link with origin 2 and destination 3, the travel distance is 45.
For the link with origin 2 and destination 4, the travel distance is 33.
For the link with origin 2 and destination 5, the travel distance is 59.
For the link with origin 2 and destination 6, the travel distance is 57.
For the link with origin 2 and destination 7, the travel distance is 70.
For the link with origin 3 and destination 1, the travel distance is 47.
For the link with origin 3 and destination 2, the travel distance is 45.
For the link with origin 3 and destination 4, the travel distance is 50.
For the link with origin 3 and destination 5, the travel distance is 62.
For the link with origin 3 and destination 6, the travel distance is 45.
For the link with origin 3 and destination 7, the travel distance is 58.
For the link with origin 4 and destination 1, the travel distance is 24.
For the link with origin 4 and destination 2, the travel distance is 33.
For the link with origin 4 and destination 3, the travel distance is 50.
For the link with origin 4 and destination 5, the travel distance is 32.
For the link with origin 4 and destination 6, the travel distance is 53.
For the link with origin 4 and destination 7, the travel distance is 66.
For the link with origin 5 and destination 1, the travel distance is 56.
For the link with origin 5 and destination 2, the travel distance is 59.
For the link with origin 5 and destination 3, the travel distance is 62.
For the link with origin 5 and destination 4, the travel distance is 32.
For the link with origin 5 and destination 6, the travel distance is 66.
For the link with origin 5 and destination 7, the travel distance is 79.
For the link with origin 6 and destination 1, the travel distance is 29.
For the link with origin 6 and destination 2, the travel distance is 57.
For the link with origin 6 and destination 3, the travel distance is 45.
For the link with origin 6 and destination 4, the travel distance is 53.
For the link with origin 6 and destination 5, the travel distance is 66.
For the link with origin 6 and destination 7, the travel distance is 13.
For the link with origin 7 and destination 1, the travel distance is 42.
For the link with origin 7 and destination 2, the travel distance is 70.
For the link with origin 7 and destination 3, the travel distance is 58.
For the link with origin 7 and destination 4, the travel distance is 66.
For the link with origin 7 and destination 5, the travel distance is 79.
For the link with origin 7 and destination 6, the travel distance is 13.
The organizer should use these distances to assign each residential sector to an opened site so the maximum assigned travel distance is minimized.

Also, when you reply with a candidate configuration, please follow this simple JSON layout so it's easy to parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

This little sketch means: ""selected"" is the list of venues you decide to open, and ""assignments"" lists, for each residential sector in the instance (in the same order they were given), which opened venue that sector is assigned to. It's just the shape I expect, not the actual answer — fill in the real identifiers from the instance when you submit.

Please make sure to use the exact identifiers from the instance input with no renaming and no new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 38, 47, 24, 56, 29, 42], [38, 0, 45, 33, 59, 57, 70], [47, 45, 0, 50, 62, 45, 58], [24, 33, 50, 0, 32, 53, 66], [56, 59, 62, 32, 0, 66, 79], [29, 57, 45, 53, 66, 0, 13], [42, 70, 58, 66, 79, 13, 0]], 'p': 1, 'objective': 56.0}","{'facilities': [0], 'assignments': [0, 0, 0, 0, 0, 0, 0]}",56.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 38, 3: 47, 4: 24, 5: 56, 6: 29, 7: 42}}, {'id': 2, 'distances': {1: 38, 2: 0, 3: 45, 4: 33, 5: 59, 6: 57, 7: 70}}, {'id': 3, 'distances': {1: 47, 2: 45, 3: 0, 4: 50, 5: 62, 6: 45, 7: 58}}, {'id': 4, 'distances': {1: 24, 2: 33, 3: 50, 4: 0, 5: 32, 6: 53, 7: 66}}, {'id': 5, 'distances': {1: 56, 2: 59, 3: 62, 4: 32, 5: 0, 6: 66, 7: 79}}, {'id': 6, 'distances': {1: 29, 2: 57, 3: 45, 4: 53, 5: 66, 6: 0, 7: 13}}, {'id': 7, 'distances': {1: 42, 2: 70, 3: 58, 4: 66, 5: 79, 6: 13, 7: 0}}], 'objective': 56.0}","{'selected': [1], 'assignments': [1, 1, 1, 1, 1, 1, 1]}",14,nl,1
PCENTER,PCENTER,"Someone on the city crew has to decide where to set up a fixed number of kiosks among several possible plazas for the upcoming festival. After choosing the plazas, they’ll assign every festival area to one of the open kiosks — every area gets exactly one kiosk and none are left out or doubled up. The practical yardstick is the longest distance any attendee would have to walk under that assignment: for each plan, look at each area’s walk to its kiosk, pick the maximum of those distances, and aim to make that maximum as small as possible. The concrete details are shown below.

# total_candidate_plazas=8
# kiosks_to_open=1
# plaza_identifiers=0, 1, 2, 3, 4, 5, 6, 7
origin_plaza_id,destination_plaza_id,walking_distance
0,1,189
0,2,220
0,3,121
0,4,176
0,5,134
0,6,204
0,7,89
1,0,189
1,2,181
1,3,139
1,4,210
1,5,81
1,6,147
1,7,123
2,0,220
2,1,181
2,3,144
2,4,280
2,5,134
2,6,200
2,7,201
3,0,121
3,1,139
3,2,144
3,4,208
3,5,58
3,6,128
3,7,127
4,0,176
4,1,210
4,2,280
4,3,208
4,5,160
4,6,90
4,7,87
5,0,134
5,1,81
5,2,134
5,3,58
5,4,160
5,6,96
5,7,114
6,0,204
6,1,147
6,2,200
6,3,128
6,4,90
6,5,96
6,7,130
7,0,89
7,1,123
7,2,201
7,3,127
7,4,87
7,5,114
7,6,130

Also, when you hand back the plan, please use this simple JSON shape so it's easy to read and check:

{
  ""solution"": {
    ""selected"": [<plaza_to_open>, <plaza_to_open>, ...],
    ""assignments"": [<assigned_plaza>, <assigned_plaza>, ...]
  }
}

Here’s what those bits mean in plain English: ""selected"" is the list of plazas you decide to open kiosks at, and ""assignments"" lists, for every festival area (in the same order as the instance), which open plaza that area is assigned to. Think of it like a short form: pick the open spots, then say which spot each area goes to.

This JSON is just a sketch of the shape I need — not the actual solution itself. Please make sure you use the exact identifiers from the instance input (no renaming, no extra labels). 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 189, 220, 121, 176, 134, 204, 89], [189, 0, 181, 139, 210, 81, 147, 123], [220, 181, 0, 144, 280, 134, 200, 201], [121, 139, 144, 0, 208, 58, 128, 127], [176, 210, 280, 208, 0, 160, 90, 87], [134, 81, 134, 58, 160, 0, 96, 114], [204, 147, 200, 128, 90, 96, 0, 130], [89, 123, 201, 127, 87, 114, 130, 0]], 'p': 1, 'objective': 160.0}","{'facilities': [5], 'assignments': [5, 5, 5, 5, 5, 5, 5, 5]}",160.0,"{'problem_type': 'PCENTER', 'num_nodes': 8, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 189, 2: 220, 3: 121, 4: 176, 5: 134, 6: 204, 7: 89}}, {'id': 1, 'distances': {0: 189, 1: 0, 2: 181, 3: 139, 4: 210, 5: 81, 6: 147, 7: 123}}, {'id': 2, 'distances': {0: 220, 1: 181, 2: 0, 3: 144, 4: 280, 5: 134, 6: 200, 7: 201}}, {'id': 3, 'distances': {0: 121, 1: 139, 2: 144, 3: 0, 4: 208, 5: 58, 6: 128, 7: 127}}, {'id': 4, 'distances': {0: 176, 1: 210, 2: 280, 3: 208, 4: 0, 5: 160, 6: 90, 7: 87}}, {'id': 5, 'distances': {0: 134, 1: 81, 2: 134, 3: 58, 4: 160, 5: 0, 6: 96, 7: 114}}, {'id': 6, 'distances': {0: 204, 1: 147, 2: 200, 3: 128, 4: 90, 5: 96, 6: 0, 7: 130}}, {'id': 7, 'distances': {0: 89, 1: 123, 2: 201, 3: 127, 4: 87, 5: 114, 6: 130, 7: 0}}], 'objective': 160.0}","{'selected': [5], 'assignments': [5, 5, 5, 5, 5, 5, 5, 5]}",15,csv,0
PCENTER,PCENTER,"We’re helping the co-op decide which farm lots should become the official collection sites: they can only open a set number of sites, and each member farm must be assigned to one and only one open site. What matters most is the farm that has the longest trip — a better choice is the one that shrinks that longest farm-to-site distance. To check a choice, calculate each farm’s distance to its assigned pickup, find the biggest distance, and aim to make that biggest number as small as practical. The exact lots and distances are shown below.

There are 6 proposed farm lots (1, 2, 3, 4, 5, 6); we must open exactly 1 collection sites from them.
We record travel from lot 1 to candidate site 2 as 36.
We record travel from lot 1 to candidate site 3 as 37.
We record travel from lot 1 to candidate site 4 as 38.
We record travel from lot 1 to candidate site 5 as 42.
We record travel from lot 1 to candidate site 6 as 33.
We record travel from lot 2 to candidate site 1 as 36.
We record travel from lot 2 to candidate site 3 as 35.
We record travel from lot 2 to candidate site 4 as 22.
We record travel from lot 2 to candidate site 5 as 40.
We record travel from lot 2 to candidate site 6 as 31.
We record travel from lot 3 to candidate site 1 as 37.
We record travel from lot 3 to candidate site 2 as 35.
We record travel from lot 3 to candidate site 4 as 33.
We record travel from lot 3 to candidate site 5 as 40.
We record travel from lot 3 to candidate site 6 as 39.
We record travel from lot 4 to candidate site 1 as 38.
We record travel from lot 4 to candidate site 2 as 22.
We record travel from lot 4 to candidate site 3 as 33.
We record travel from lot 4 to candidate site 5 as 28.
We record travel from lot 4 to candidate site 6 as 24.
We record travel from lot 5 to candidate site 1 as 42.
We record travel from lot 5 to candidate site 2 as 40.
We record travel from lot 5 to candidate site 3 as 40.
We record travel from lot 5 to candidate site 4 as 28.
We record travel from lot 5 to candidate site 6 as 35.
We record travel from lot 6 to candidate site 1 as 33.
We record travel from lot 6 to candidate site 2 as 31.
We record travel from lot 6 to candidate site 3 as 39.
We record travel from lot 6 to candidate site 4 as 24.
We record travel from lot 6 to candidate site 5 as 35.
Use these entries to compute each farm’s assigned distance and the maximum (worst-case) travel so we can pick the best 1 sites for the co-op.

Also, to keep things machine-friendly (and simple for us), please return your choice in this little JSON shape when you're ready — nothing fancy, just the keys and lists:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

Here’s what those bits mean in plain English: ""selected"" is the list of lot identifiers you want to open as pickup sites, and ""assignments"" is a list (in the same order as the farms were given in the input) saying which open site each farm will use. Think of it like filling out a short form: which sites are open, and for each farm, which open site it's assigned to.

This is just a sketch of the shape I need, not the actual answer. One important note: use the exact identifiers from the instance input — don’t rename them and don’t invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 36, 37, 38, 42, 33], [36, 0, 35, 22, 40, 31], [37, 35, 0, 33, 40, 39], [38, 22, 33, 0, 28, 24], [42, 40, 40, 28, 0, 35], [33, 31, 39, 24, 35, 0]], 'p': 1, 'objective': 38.0}","{'facilities': [3], 'assignments': [3, 3, 3, 3, 3, 3]}",38.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 36, 3: 37, 4: 38, 5: 42, 6: 33}}, {'id': 2, 'distances': {1: 36, 2: 0, 3: 35, 4: 22, 5: 40, 6: 31}}, {'id': 3, 'distances': {1: 37, 2: 35, 3: 0, 4: 33, 5: 40, 6: 39}}, {'id': 4, 'distances': {1: 38, 2: 22, 3: 33, 4: 0, 5: 28, 6: 24}}, {'id': 5, 'distances': {1: 42, 2: 40, 3: 40, 4: 28, 5: 0, 6: 35}}, {'id': 6, 'distances': {1: 33, 2: 31, 3: 39, 4: 24, 5: 35, 6: 0}}], 'objective': 38.0}","{'selected': [4], 'assignments': [4, 4, 4, 4, 4, 4]}",16,nl,1
PCENTER,PCENTER,"Many people might think this is a routing puzzle, but it’s really about picking a fixed number of storefronts to open as micro post offices and assigning every delivery cluster to one opened office. Every location must belong to one and only one office, and the selection is better when the longest trip any cluster must make to its assigned office is reduced. Practically speaking, check all cluster-to-office distances in a plan and the biggest of those distances is the score to beat — lower is better. See the exact candidate storefronts and the travel distances below.

# total_locations=7
# num_micro_post_offices=2
# location_ids=0, 1, 2, 3, 4, 5, 6
source_location_id,target_location_id,travel_distance
0,1,47
0,2,45
0,3,12
0,4,45
0,5,37
0,6,39
1,0,47
1,2,37
1,3,45
1,4,20
1,5,33
1,6,38
2,0,45
2,1,37
2,3,41
2,4,45
2,5,23
2,6,39
3,0,12
3,1,45
3,2,41
3,4,41
3,5,33
3,6,27
4,0,45
4,1,20
4,2,45
4,3,41
4,5,38
4,6,40
5,0,37
5,1,33
5,2,23
5,3,33
5,4,38
5,6,29
6,0,39
6,1,38
6,2,39
6,3,27
6,4,40
6,5,29

Also, when you reply, please follow this simple JSON layout so I can read your plan easily—here’s the sketch of the shape I expect:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

This just means: put the storefronts you want to open in the ""selected"" list, and then list, for every input location (in the same order as the instance), the opened storefront it gets assigned to in ""assignments"". Think of it like a short form: the first part says which offices will be open, the second part says where each neighborhood goes. Super informal — just fill those placeholders with the actual identifiers from the instance when you answer.

Quick reminder: the JSON above is only a template of the expected shape, not the final answer. All identifiers you use in your real answer must match the instance input exactly — do not rename them or invent new labels. Valid identifiers look like:
- plain numbers such as ""1"" or ""23""
- single capital letters like ""A"" or ""B""
- a capital letter followed by digits like ""A1"" or ""X7""","{'distance_matrix': [[0, 47, 45, 12, 45, 37, 39], [47, 0, 37, 45, 20, 33, 38], [45, 37, 0, 41, 45, 23, 39], [12, 45, 41, 0, 41, 33, 27], [45, 20, 45, 41, 0, 38, 40], [37, 33, 23, 33, 38, 0, 29], [39, 38, 39, 27, 40, 29, 0]], 'p': 2, 'objective': 37.0}","{'facilities': [4, 5], 'assignments': [5, 5, 5, 5, 4, 5, 5]}",37.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 2, 'sites': [{'id': 0, 'distances': {0: 0, 1: 47, 2: 45, 3: 12, 4: 45, 5: 37, 6: 39}}, {'id': 1, 'distances': {0: 47, 1: 0, 2: 37, 3: 45, 4: 20, 5: 33, 6: 38}}, {'id': 2, 'distances': {0: 45, 1: 37, 2: 0, 3: 41, 4: 45, 5: 23, 6: 39}}, {'id': 3, 'distances': {0: 12, 1: 45, 2: 41, 3: 0, 4: 41, 5: 33, 6: 27}}, {'id': 4, 'distances': {0: 45, 1: 20, 2: 45, 3: 41, 4: 0, 5: 38, 6: 40}}, {'id': 5, 'distances': {0: 37, 1: 33, 2: 23, 3: 33, 4: 38, 5: 0, 6: 29}}, {'id': 6, 'distances': {0: 39, 1: 38, 2: 39, 3: 27, 4: 40, 5: 29, 6: 0}}], 'objective': 37.0}","{'selected': [4, 5], 'assignments': [5, 5, 5, 5, 4, 5, 5]}",17,csv,0
PCENTER,PCENTER,"Many people on the trail team are talking about a simple task: pick a set number of box sites out of a list, open them, and assign each piece of trail to a single open box. The goal is to improve the worst-off hiker — check every trail segment’s walk to its box, take the maximum of those walks, and try to make that maximum as small as possible. All the candidate sites and the exact distances between points are detailed below.

{
  ""total_candidate_sites"": 7,
  ""boxes_to_install"": 1,
  ""site_ids"": [
    0,
    1,
    2,
    3,
    4,
    5,
    6
  ],
  ""data"": [
    {
      ""from_site_id"": 0,
      ""to_site_id"": 1,
      ""walking_distance_between_sites"": 42
    },
    {
      ""from_site_id"": 0,
      ""to_site_id"": 2,
      ""walking_distance_between_sites"": 49
    },
    {
      ""from_site_id"": 0,
      ""to_site_id"": 3,
      ""walking_distance_between_sites"": 97
    },
    {
      ""from_site_id"": 0,
      ""to_site_id"": 4,
      ""walking_distance_between_sites"": 151
    },
    {
      ""from_site_id"": 0,
      ""to_site_id"": 5,
      ""walking_distance_between_sites"": 157
    },
    {
      ""from_site_id"": 0,
      ""to_site_id"": 6,
      ""walking_distance_between_sites"": 154
    },
    {
      ""from_site_id"": 1,
      ""to_site_id"": 0,
      ""walking_distance_between_sites"": 42
    },
    {
      ""from_site_id"": 1,
      ""to_site_id"": 2,
      ""walking_distance_between_sites"": 91
    },
    {
      ""from_site_id"": 1,
      ""to_site_id"": 3,
      ""walking_distance_between_sites"": 55
    },
    {
      ""from_site_id"": 1,
      ""to_site_id"": 4,
      ""walking_distance_between_sites"": 109
    },
    {
      ""from_site_id"": 1,
      ""to_site_id"": 5,
      ""walking_distance_between_sites"": 147
    },
    {
      ""from_site_id"": 1,
      ""to_site_id"": 6,
      ""walking_distance_between_sites"": 134
    },
    {
      ""from_site_id"": 2,
      ""to_site_id"": 0,
      ""walking_distance_between_sites"": 49
    },
    {
      ""from_site_id"": 2,
      ""to_site_id"": 1,
      ""walking_distance_between_sites"": 91
    },
    {
      ""from_site_id"": 2,
      ""to_site_id"": 3,
      ""walking_distance_between_sites"": 133
    },
    {
      ""from_site_id"": 2,
      ""to_site_id"": 4,
      ""walking_distance_between_sites"": 139
    },
    {
      ""from_site_id"": 2,
      ""to_site_id"": 5,
      ""walking_distance_between_sites"": 194
    },
    {
      ""from_site_id"": 2,
      ""to_site_id"": 6,
      ""walking_distance_between_sites"": 203
    },
    {
      ""from_site_id"": 3,
      ""to_site_id"": 0,
      ""walking_distance_between_sites"": 97
    },
    {
      ""from_site_id"": 3,
      ""to_site_id"": 1,
      ""walking_distance_between_sites"": 55
    },
    {
      ""from_site_id"": 3,
      ""to_site_id"": 2,
      ""walking_distance_between_sites"": 133
    },
    {
      ""from_site_id"": 3,
      ""to_site_id"": 4,
      ""walking_distance_between_sites"": 54
    },
    {
      ""from_site_id"": 3,
      ""to_site_id"": 5,
      ""walking_distance_between_sites"": 162
    },
    {
      ""from_site_id"": 3,
      ""to_site_id"": 6,
      ""walking_distance_between_sites"": 182
    },
    {
      ""from_site_id"": 4,
      ""to_site_id"": 0,
      ""walking_distance_between_sites"": 151
    },
    {
      ""from_site_id"": 4,
      ""to_site_id"": 1,
      ""walking_distance_between_sites"": 109
    },
    {
      ""from_site_id"": 4,
      ""to_site_id"": 2,
      ""walking_distance_between_sites"": 139
    },
    {
      ""from_site_id"": 4,
      ""to_site_id"": 3,
      ""walking_distance_between_sites"": 54
    },
    {
      ""from_site_id"": 4,
      ""to_site_id"": 5,
      ""walking_distance_between_sites"": 139
    },
    {
      ""from_site_id"": 4,
      ""to_site_id"": 6,
      ""walking_distance_between_sites"": 182
    },
    {
      ""from_site_id"": 5,
      ""to_site_id"": 0,
      ""walking_distance_between_sites"": 157
    },
    {
      ""from_site_id"": 5,
      ""to_site_id"": 1,
      ""walking_distance_between_sites"": 147
    },
    {
      ""from_site_id"": 5,
      ""to_site_id"": 2,
      ""walking_distance_between_sites"": 194
    },
    {
      ""from_site_id"": 5,
      ""to_site_id"": 3,
      ""walking_distance_between_sites"": 162
    },
    {
      ""from_site_id"": 5,
      ""to_site_id"": 4,
      ""walking_distance_between_sites"": 139
    },
    {
      ""from_site_id"": 5,
      ""to_site_id"": 6,
      ""walking_distance_between_sites"": 211
    },
    {
      ""from_site_id"": 6,
      ""to_site_id"": 0,
      ""walking_distance_between_sites"": 154
    },
    {
      ""from_site_id"": 6,
      ""to_site_id"": 1,
      ""walking_distance_between_sites"": 134
    },
    {
      ""from_site_id"": 6,
      ""to_site_id"": 2,
      ""walking_distance_between_sites"": 203
    },
    {
      ""from_site_id"": 6,
      ""to_site_id"": 3,
      ""walking_distance_between_sites"": 182
    },
    {
      ""from_site_id"": 6,
      ""to_site_id"": 4,
      ""walking_distance_between_sites"": 182
    },
    {
      ""from_site_id"": 6,
      ""to_site_id"": 5,
      ""walking_distance_between_sites"": 211
    }
  ]
}

Oh, and to keep things machine-friendly, please shape your answer like this simple JSON snippet:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

Think of ""selected"" as the list of box sites you decide to open, and ""assignments"" as which open site each trail segment is assigned to. Super casual: one list says which boxes are open, the other says where every piece of trail goes. This is just the expected shape — not the real answer itself.

Also, a quick reminder: use the exact identifiers from the instance input — do not rename them and do not invent new labels. For example: Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 42, 49, 97, 151, 157, 154], [42, 0, 91, 55, 109, 147, 134], [49, 91, 0, 133, 139, 194, 203], [97, 55, 133, 0, 54, 162, 182], [151, 109, 139, 54, 0, 139, 182], [157, 147, 194, 162, 139, 0, 211], [154, 134, 203, 182, 182, 211, 0]], 'p': 1, 'objective': 147.0}","{'facilities': [1], 'assignments': [1, 1, 1, 1, 1, 1, 1]}",147.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 42, 2: 49, 3: 97, 4: 151, 5: 157, 6: 154}}, {'id': 1, 'distances': {0: 42, 1: 0, 2: 91, 3: 55, 4: 109, 5: 147, 6: 134}}, {'id': 2, 'distances': {0: 49, 1: 91, 2: 0, 3: 133, 4: 139, 5: 194, 6: 203}}, {'id': 3, 'distances': {0: 97, 1: 55, 2: 133, 3: 0, 4: 54, 5: 162, 6: 182}}, {'id': 4, 'distances': {0: 151, 1: 109, 2: 139, 3: 54, 4: 0, 5: 139, 6: 182}}, {'id': 5, 'distances': {0: 157, 1: 147, 2: 194, 3: 162, 4: 139, 5: 0, 6: 211}}, {'id': 6, 'distances': {0: 154, 1: 134, 2: 203, 3: 182, 4: 182, 5: 211, 6: 0}}], 'objective': 147.0}","{'selected': [1], 'assignments': [1, 1, 1, 1, 1, 1, 1]}",18,json,0
PCENTER,PCENTER,"In town there’s a push to place a limited number of pop-up galleries at selected venues; the task is to decide which ones to open and then attach every neighborhood arts group to exactly one of those opened galleries so no group is forgotten or doubled up. The way to judge a choice is by its worst-case trip: calculate each group’s travel to its assigned gallery, find the group with the longest trip, and try to make that longest trip as short as possible. The specific list of venues, the travel distances, and the fixed number of pop-ups available are shown below.

Below are the 5 candidate locations, the fixed 2 pop-ups to open, and the location IDs: 0, 1, 2, 3, 4.
Travel distance from 0 to 1 is 33.
Travel distance from 0 to 2 is 27.
Travel distance from 0 to 3 is 31.
Travel distance from 0 to 4 is 29.
Travel distance from 1 to 0 is 33.
Travel distance from 1 to 2 is 30.
Travel distance from 1 to 3 is 40.
Travel distance from 1 to 4 is 13.
Travel distance from 2 to 0 is 27.
Travel distance from 2 to 1 is 30.
Travel distance from 2 to 3 is 29.
Travel distance from 2 to 4 is 32.
Travel distance from 3 to 0 is 31.
Travel distance from 3 to 1 is 40.
Travel distance from 3 to 2 is 29.
Travel distance from 3 to 4 is 29.
Travel distance from 4 to 0 is 29.
Travel distance from 4 to 1 is 13.
Travel distance from 4 to 2 is 32.
Travel distance from 4 to 3 is 29.
Use these distances to assign each group and minimize the worst-case trip among the 5 locations with 2 pop-ups.

You can give the result back in a tiny JSON form so it's easy to read and parse. Something like this is the shape I expect:

{
  ""solution"": {
    ""selected"": [<venue_to_open>, <venue_to_open>, ...],
    ""assignments"": [<assigned_open_venue>, <assigned_open_venue>, ...]
  }
}

Here ""selected"" is where you list the venues you'll open (the pop-up spots), and ""assignments"" is a one-to-one list saying which opened venue each neighborhood arts group is attached to. Think of it like filling out a short form: pick the open galleries, then for every group write which open gallery they go to.

This is just the required shape — a sketch, not the actual answer filled in.

Please use the exact identifiers from the instance input when you fill this in — do not rename them or invent new labels. For example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 33, 27, 31, 29], [33, 0, 30, 40, 13], [27, 30, 0, 29, 32], [31, 40, 29, 0, 29], [29, 13, 32, 29, 0]], 'p': 2, 'objective': 29.0}","{'facilities': [1, 2], 'assignments': [2, 1, 2, 2, 1]}",29.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 2, 'sites': [{'id': 0, 'distances': {0: 0, 1: 33, 2: 27, 3: 31, 4: 29}}, {'id': 1, 'distances': {0: 33, 1: 0, 2: 30, 3: 40, 4: 13}}, {'id': 2, 'distances': {0: 27, 1: 30, 2: 0, 3: 29, 4: 32}}, {'id': 3, 'distances': {0: 31, 1: 40, 2: 29, 3: 0, 4: 29}}, {'id': 4, 'distances': {0: 29, 1: 13, 2: 32, 3: 29, 4: 0}}], 'objective': 29.0}","{'selected': [1, 2], 'assignments': [2, 1, 2, 2, 1]}",19,nl,0
PCENTER,PCENTER,"We ran into a common logistics puzzle: the ride-share ops manager must pick a set number of staging spots from candidate road segments and then link each service area to exactly one of the opened spots. Every area has to be assigned and nobody can be assigned twice. One plan is better than another if the farthest any area has to travel to its assigned spot is smaller. To evaluate a plan, measure the distance from each area to its assigned staging zone and take the maximum of those distances — that maximum is what the team wants to minimize. The specific inputs and distances are listed below.

{
  ""num_candidate_road_segments"": 7,
  ""num_staging_zones_to_open"": 1,
  ""road_segment_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G""
  ],
  ""data"": [
    {
      ""from_segment_id"": ""A"",
      ""to_segment_id"": ""B"",
      ""repositioning_distance"": 85
    },
    {
      ""from_segment_id"": ""A"",
      ""to_segment_id"": ""C"",
      ""repositioning_distance"": 55
    },
    {
      ""from_segment_id"": ""A"",
      ""to_segment_id"": ""D"",
      ""repositioning_distance"": 97
    },
    {
      ""from_segment_id"": ""A"",
      ""to_segment_id"": ""E"",
      ""repositioning_distance"": 63
    },
    {
      ""from_segment_id"": ""A"",
      ""to_segment_id"": ""F"",
      ""repositioning_distance"": 76
    },
    {
      ""from_segment_id"": ""A"",
      ""to_segment_id"": ""G"",
      ""repositioning_distance"": 100
    },
    {
      ""from_segment_id"": ""B"",
      ""to_segment_id"": ""A"",
      ""repositioning_distance"": 85
    },
    {
      ""from_segment_id"": ""B"",
      ""to_segment_id"": ""C"",
      ""repositioning_distance"": 62
    },
    {
      ""from_segment_id"": ""B"",
      ""to_segment_id"": ""D"",
      ""repositioning_distance"": 112
    },
    {
      ""from_segment_id"": ""B"",
      ""to_segment_id"": ""E"",
      ""repositioning_distance"": 44
    },
    {
      ""from_segment_id"": ""B"",
      ""to_segment_id"": ""F"",
      ""repositioning_distance"": 71
    },
    {
      ""from_segment_id"": ""B"",
      ""to_segment_id"": ""G"",
      ""repositioning_distance"": 94
    },
    {
      ""from_segment_id"": ""C"",
      ""to_segment_id"": ""A"",
      ""repositioning_distance"": 55
    },
    {
      ""from_segment_id"": ""C"",
      ""to_segment_id"": ""B"",
      ""repositioning_distance"": 62
    },
    {
      ""from_segment_id"": ""C"",
      ""to_segment_id"": ""D"",
      ""repositioning_distance"": 86
    },
    {
      ""from_segment_id"": ""C"",
      ""to_segment_id"": ""E"",
      ""repositioning_distance"": 68
    },
    {
      ""from_segment_id"": ""C"",
      ""to_segment_id"": ""F"",
      ""repositioning_distance"": 66
    },
    {
      ""from_segment_id"": ""C"",
      ""to_segment_id"": ""G"",
      ""repositioning_distance"": 83
    },
    {
      ""from_segment_id"": ""D"",
      ""to_segment_id"": ""A"",
      ""repositioning_distance"": 97
    },
    {
      ""from_segment_id"": ""D"",
      ""to_segment_id"": ""B"",
      ""repositioning_distance"": 112
    },
    {
      ""from_segment_id"": ""D"",
      ""to_segment_id"": ""C"",
      ""repositioning_distance"": 86
    },
    {
      ""from_segment_id"": ""D"",
      ""to_segment_id"": ""E"",
      ""repositioning_distance"": 68
    },
    {
      ""from_segment_id"": ""D"",
      ""to_segment_id"": ""F"",
      ""repositioning_distance"": 55
    },
    {
      ""from_segment_id"": ""D"",
      ""to_segment_id"": ""G"",
      ""repositioning_distance"": 85
    },
    {
      ""from_segment_id"": ""E"",
      ""to_segment_id"": ""A"",
      ""repositioning_distance"": 63
    },
    {
      ""from_segment_id"": ""E"",
      ""to_segment_id"": ""B"",
      ""repositioning_distance"": 44
    },
    {
      ""from_segment_id"": ""E"",
      ""to_segment_id"": ""C"",
      ""repositioning_distance"": 68
    },
    {
      ""from_segment_id"": ""E"",
      ""to_segment_id"": ""D"",
      ""repositioning_distance"": 68
    },
    {
      ""from_segment_id"": ""E"",
      ""to_segment_id"": ""F"",
      ""repositioning_distance"": 27
    },
    {
      ""from_segment_id"": ""E"",
      ""to_segment_id"": ""G"",
      ""repositioning_distance"": 71
    },
    {
      ""from_segment_id"": ""F"",
      ""to_segment_id"": ""A"",
      ""repositioning_distance"": 76
    },
    {
      ""from_segment_id"": ""F"",
      ""to_segment_id"": ""B"",
      ""repositioning_distance"": 71
    },
    {
      ""from_segment_id"": ""F"",
      ""to_segment_id"": ""C"",
      ""repositioning_distance"": 66
    },
    {
      ""from_segment_id"": ""F"",
      ""to_segment_id"": ""D"",
      ""repositioning_distance"": 55
    },
    {
      ""from_segment_id"": ""F"",
      ""to_segment_id"": ""E"",
      ""repositioning_distance"": 27
    },
    {
      ""from_segment_id"": ""F"",
      ""to_segment_id"": ""G"",
      ""repositioning_distance"": 69
    },
    {
      ""from_segment_id"": ""G"",
      ""to_segment_id"": ""A"",
      ""repositioning_distance"": 100
    },
    {
      ""from_segment_id"": ""G"",
      ""to_segment_id"": ""B"",
      ""repositioning_distance"": 94
    },
    {
      ""from_segment_id"": ""G"",
      ""to_segment_id"": ""C"",
      ""repositioning_distance"": 83
    },
    {
      ""from_segment_id"": ""G"",
      ""to_segment_id"": ""D"",
      ""repositioning_distance"": 85
    },
    {
      ""from_segment_id"": ""G"",
      ""to_segment_id"": ""E"",
      ""repositioning_distance"": 71
    },
    {
      ""from_segment_id"": ""G"",
      ""to_segment_id"": ""F"",
      ""repositioning_distance"": 69
    }
  ]
}

If you want to hand back a candidate plan, just use a simple JSON shape like this — nothing fancy, just a small form that tells us which spots to open and how each area gets assigned:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_spot>, <chosen_open_spot>, ...]
  }
}

Think of ""selected"" as the list of staging spots you decide to open, and ""assignments"" as, for each service area (in the same order as the input), which opened spot it will use. This is just a sketch of the shape I need, not the actual answer — replace the angle-bracket placeholders with the real identifiers from the instance.

Please make sure all identifiers are used exactly as they appear in the instance input — no renaming and no new labels.
- for example: Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 85, 55, 97, 63, 76, 100], [85, 0, 62, 112, 44, 71, 94], [55, 62, 0, 86, 68, 66, 83], [97, 112, 86, 0, 68, 55, 85], [63, 44, 68, 68, 0, 27, 71], [76, 71, 66, 55, 27, 0, 69], [100, 94, 83, 85, 71, 69, 0]], 'p': 1, 'objective': 71.0}","{'facilities': [4], 'assignments': [4, 4, 4, 4, 4, 4, 4]}",71.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 85, 'C': 55, 'D': 97, 'E': 63, 'F': 76, 'G': 100}}, {'id': 'B', 'distances': {'A': 85, 'B': 0, 'C': 62, 'D': 112, 'E': 44, 'F': 71, 'G': 94}}, {'id': 'C', 'distances': {'A': 55, 'B': 62, 'C': 0, 'D': 86, 'E': 68, 'F': 66, 'G': 83}}, {'id': 'D', 'distances': {'A': 97, 'B': 112, 'C': 86, 'D': 0, 'E': 68, 'F': 55, 'G': 85}}, {'id': 'E', 'distances': {'A': 63, 'B': 44, 'C': 68, 'D': 68, 'E': 0, 'F': 27, 'G': 71}}, {'id': 'F', 'distances': {'A': 76, 'B': 71, 'C': 66, 'D': 55, 'E': 27, 'F': 0, 'G': 69}}, {'id': 'G', 'distances': {'A': 100, 'B': 94, 'C': 83, 'D': 85, 'E': 71, 'F': 69, 'G': 0}}], 'objective': 71.0}","{'selected': ['E'], 'assignments': ['E', 'E', 'E', 'E', 'E', 'E', 'E']}",20,json,names
PCENTER,PCENTER,"Someone running the festival needs to pick a set number of water points from a list of candidate sites, and then link each gathering zone to one of those water points. The quality of a plan is judged by the worst walk anyone faces: for every zone find its walk to its linked point, then the largest of those walks is the value to reduce. Nothing can be left out or doubly linked — every zone gets one and only one connection, and chosen water points come from the available spots. The concrete layout and distances appear below.

# total_candidate_locations=6
# water_stations_to_open=2
# location_ids=0, 1, 2, 3, 4, 5
source_location_id,target_location_id,walking_distance_between_locations
0,1,53
0,2,43
0,3,35
0,4,48
0,5,48
1,0,53
1,2,25
1,3,33
1,4,28
1,5,40
2,0,43
2,1,25
2,3,20
2,4,26
2,5,25
3,0,35
3,1,33
3,2,20
3,4,29
3,5,25
4,0,48
4,1,28
4,2,26
4,3,29
4,5,37
5,0,48
5,1,40
5,2,25
5,3,25
5,4,37

You can reply using this simple JSON shape so it's easy to read and check automatically. Here's the sketch of how I expect it to look:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

""selected"" is where you list the water-point sites you decide to open. ""assignments"" is a list that, for every gathering zone in the order the instance gives them, names the open site that zone is linked to. Think of it like a short form: which sites are open, then for each zone which open site it uses.

This is just the shape I need — fill in the real identifiers from the instance when you answer, don't invent anything. No renaming of identifiers and no new labels allowed.

- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 53, 43, 35, 48, 48], [53, 0, 25, 33, 28, 40], [43, 25, 0, 20, 26, 25], [35, 33, 20, 0, 29, 25], [48, 28, 26, 29, 0, 37], [48, 40, 25, 25, 37, 0]], 'p': 2, 'objective': 26.0}","{'facilities': [0, 2], 'assignments': [0, 2, 2, 2, 2, 2]}",26.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 2, 'sites': [{'id': 0, 'distances': {0: 0, 1: 53, 2: 43, 3: 35, 4: 48, 5: 48}}, {'id': 1, 'distances': {0: 53, 1: 0, 2: 25, 3: 33, 4: 28, 5: 40}}, {'id': 2, 'distances': {0: 43, 1: 25, 2: 0, 3: 20, 4: 26, 5: 25}}, {'id': 3, 'distances': {0: 35, 1: 33, 2: 20, 3: 0, 4: 29, 5: 25}}, {'id': 4, 'distances': {0: 48, 1: 28, 2: 26, 3: 29, 4: 0, 5: 37}}, {'id': 5, 'distances': {0: 48, 1: 40, 2: 25, 3: 25, 4: 37, 5: 0}}], 'objective': 26.0}","{'selected': [0, 2], 'assignments': [0, 2, 2, 2, 2, 2]}",21,csv,0
PCENTER,PCENTER,"There’s a company trying to decide which sites to activate as depots — but only a fixed number can be opened — and every delivery point has to be tied to exactly one open depot, no exceptions. Success is judged by the longest trip any delivery point has to take: add up nothing, just find the biggest distance among all assignments, and the better plan is the one with the smaller biggest-distance. The concrete locations, pairwise distances, and the number of depots to choose appear below.

The instance lists 7 candidate sites: 1, 2, 3, 4, 5, 6, 7. The company must open exactly 1 depots.
The travel distance from 1 to 2 is 38.
The travel distance from 1 to 3 is 45.
The travel distance from 1 to 4 is 38.
The travel distance from 1 to 5 is 48.
The travel distance from 1 to 6 is 43.
The travel distance from 1 to 7 is 41.
The travel distance from 2 to 1 is 38.
The travel distance from 2 to 3 is 23.
The travel distance from 2 to 4 is 32.
The travel distance from 2 to 5 is 35.
The travel distance from 2 to 6 is 28.
The travel distance from 2 to 7 is 19.
The travel distance from 3 to 1 is 45.
The travel distance from 3 to 2 is 23.
The travel distance from 3 to 4 is 39.
The travel distance from 3 to 5 is 39.
The travel distance from 3 to 6 is 29.
The travel distance from 3 to 7 is 20.
The travel distance from 4 to 1 is 38.
The travel distance from 4 to 2 is 32.
The travel distance from 4 to 3 is 39.
The travel distance from 4 to 5 is 37.
The travel distance from 4 to 6 is 40.
The travel distance from 4 to 7 is 35.
The travel distance from 5 to 1 is 48.
The travel distance from 5 to 2 is 35.
The travel distance from 5 to 3 is 39.
The travel distance from 5 to 4 is 37.
The travel distance from 5 to 6 is 29.
The travel distance from 5 to 7 is 33.
The travel distance from 6 to 1 is 43.
The travel distance from 6 to 2 is 28.
The travel distance from 6 to 3 is 29.
The travel distance from 6 to 4 is 40.
The travel distance from 6 to 5 is 29.
The travel distance from 6 to 7 is 25.
The travel distance from 7 to 1 is 41.
The travel distance from 7 to 2 is 19.
The travel distance from 7 to 3 is 20.
The travel distance from 7 to 4 is 35.
The travel distance from 7 to 5 is 33.
The travel distance from 7 to 6 is 25.
All listed pairwise distances follow above; choose 1 depots so every one of the 7 sites is assigned and the company minimizes the maximum assigned distance.

Also, when you send the actual plan back, please stick to this simple JSON layout so it's easy to parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_site>, <assigned_site>, ...]
  }
}

Think of ""selected"" as the list of sites you decide to activate, and ""assignments"" as which open site each delivery point is tied to (one entry per location in the same order as the instance). Super casual: ""selected"" = where we open depots, ""assignments"" = which depot each place uses.

This JSON is just a sketch of the shape I need, not the real answer.

Please make sure to use the exact identifiers given in the instance input — do not rename them or invent new labels.  
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 38, 45, 38, 48, 43, 41], [38, 0, 23, 32, 35, 28, 19], [45, 23, 0, 39, 39, 29, 20], [38, 32, 39, 0, 37, 40, 35], [48, 35, 39, 37, 0, 29, 33], [43, 28, 29, 40, 29, 0, 25], [41, 19, 20, 35, 33, 25, 0]], 'p': 1, 'objective': 38.0}","{'facilities': [1], 'assignments': [1, 1, 1, 1, 1, 1, 1]}",38.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 38, 3: 45, 4: 38, 5: 48, 6: 43, 7: 41}}, {'id': 2, 'distances': {1: 38, 2: 0, 3: 23, 4: 32, 5: 35, 6: 28, 7: 19}}, {'id': 3, 'distances': {1: 45, 2: 23, 3: 0, 4: 39, 5: 39, 6: 29, 7: 20}}, {'id': 4, 'distances': {1: 38, 2: 32, 3: 39, 4: 0, 5: 37, 6: 40, 7: 35}}, {'id': 5, 'distances': {1: 48, 2: 35, 3: 39, 4: 37, 5: 0, 6: 29, 7: 33}}, {'id': 6, 'distances': {1: 43, 2: 28, 3: 29, 4: 40, 5: 29, 6: 0, 7: 25}}, {'id': 7, 'distances': {1: 41, 2: 19, 3: 20, 4: 35, 5: 33, 6: 25, 7: 0}}], 'objective': 38.0}","{'selected': [2], 'assignments': [2, 2, 2, 2, 2, 2, 2]}",22,markdown_table,1
PCENTER,PCENTER,"Recently the neighborhood decided they’ll add a set number of seating areas drawn from several candidate sites, and the job is to choose which ones and assign each street segment to one opened seating area — each segment must have one, and only one, assigned spot. To see which choice works best, compute the distance from every street segment to its assigned seat, then take the biggest of those values; the aim is to make that biggest distance as small as it can be. The exact list of candidate spots and distances follows below.

{
  ""total_locations_count"": 7,
  ""seating_areas_to_install"": 1,
  ""location_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G""
  ],
  ""data"": [
    {
      ""from_location_id"": ""A"",
      ""to_location_id"": ""B"",
      ""walking_distance"": 25
    },
    {
      ""from_location_id"": ""A"",
      ""to_location_id"": ""C"",
      ""walking_distance"": 29
    },
    {
      ""from_location_id"": ""A"",
      ""to_location_id"": ""D"",
      ""walking_distance"": 41
    },
    {
      ""from_location_id"": ""A"",
      ""to_location_id"": ""E"",
      ""walking_distance"": 49
    },
    {
      ""from_location_id"": ""A"",
      ""to_location_id"": ""F"",
      ""walking_distance"": 49
    },
    {
      ""from_location_id"": ""A"",
      ""to_location_id"": ""G"",
      ""walking_distance"": 39
    },
    {
      ""from_location_id"": ""B"",
      ""to_location_id"": ""A"",
      ""walking_distance"": 25
    },
    {
      ""from_location_id"": ""B"",
      ""to_location_id"": ""C"",
      ""walking_distance"": 36
    },
    {
      ""from_location_id"": ""B"",
      ""to_location_id"": ""D"",
      ""walking_distance"": 24
    },
    {
      ""from_location_id"": ""B"",
      ""to_location_id"": ""E"",
      ""walking_distance"": 54
    },
    {
      ""from_location_id"": ""B"",
      ""to_location_id"": ""F"",
      ""walking_distance"": 52
    },
    {
      ""from_location_id"": ""B"",
      ""to_location_id"": ""G"",
      ""walking_distance"": 38
    },
    {
      ""from_location_id"": ""C"",
      ""to_location_id"": ""A"",
      ""walking_distance"": 29
    },
    {
      ""from_location_id"": ""C"",
      ""to_location_id"": ""B"",
      ""walking_distance"": 36
    },
    {
      ""from_location_id"": ""C"",
      ""to_location_id"": ""D"",
      ""walking_distance"": 27
    },
    {
      ""from_location_id"": ""C"",
      ""to_location_id"": ""E"",
      ""walking_distance"": 58
    },
    {
      ""from_location_id"": ""C"",
      ""to_location_id"": ""F"",
      ""walking_distance"": 59
    },
    {
      ""from_location_id"": ""C"",
      ""to_location_id"": ""G"",
      ""walking_distance"": 52
    },
    {
      ""from_location_id"": ""D"",
      ""to_location_id"": ""A"",
      ""walking_distance"": 41
    },
    {
      ""from_location_id"": ""D"",
      ""to_location_id"": ""B"",
      ""walking_distance"": 24
    },
    {
      ""from_location_id"": ""D"",
      ""to_location_id"": ""C"",
      ""walking_distance"": 27
    },
    {
      ""from_location_id"": ""D"",
      ""to_location_id"": ""E"",
      ""walking_distance"": 53
    },
    {
      ""from_location_id"": ""D"",
      ""to_location_id"": ""F"",
      ""walking_distance"": 70
    },
    {
      ""from_location_id"": ""D"",
      ""to_location_id"": ""G"",
      ""walking_distance"": 49
    },
    {
      ""from_location_id"": ""E"",
      ""to_location_id"": ""A"",
      ""walking_distance"": 49
    },
    {
      ""from_location_id"": ""E"",
      ""to_location_id"": ""B"",
      ""walking_distance"": 54
    },
    {
      ""from_location_id"": ""E"",
      ""to_location_id"": ""C"",
      ""walking_distance"": 58
    },
    {
      ""from_location_id"": ""E"",
      ""to_location_id"": ""D"",
      ""walking_distance"": 53
    },
    {
      ""from_location_id"": ""E"",
      ""to_location_id"": ""F"",
      ""walking_distance"": 77
    },
    {
      ""from_location_id"": ""E"",
      ""to_location_id"": ""G"",
      ""walking_distance"": 61
    },
    {
      ""from_location_id"": ""F"",
      ""to_location_id"": ""A"",
      ""walking_distance"": 49
    },
    {
      ""from_location_id"": ""F"",
      ""to_location_id"": ""B"",
      ""walking_distance"": 52
    },
    {
      ""from_location_id"": ""F"",
      ""to_location_id"": ""C"",
      ""walking_distance"": 59
    },
    {
      ""from_location_id"": ""F"",
      ""to_location_id"": ""D"",
      ""walking_distance"": 70
    },
    {
      ""from_location_id"": ""F"",
      ""to_location_id"": ""E"",
      ""walking_distance"": 77
    },
    {
      ""from_location_id"": ""F"",
      ""to_location_id"": ""G"",
      ""walking_distance"": 65
    },
    {
      ""from_location_id"": ""G"",
      ""to_location_id"": ""A"",
      ""walking_distance"": 39
    },
    {
      ""from_location_id"": ""G"",
      ""to_location_id"": ""B"",
      ""walking_distance"": 38
    },
    {
      ""from_location_id"": ""G"",
      ""to_location_id"": ""C"",
      ""walking_distance"": 52
    },
    {
      ""from_location_id"": ""G"",
      ""to_location_id"": ""D"",
      ""walking_distance"": 49
    },
    {
      ""from_location_id"": ""G"",
      ""to_location_id"": ""E"",
      ""walking_distance"": 61
    },
    {
      ""from_location_id"": ""G"",
      ""to_location_id"": ""F"",
      ""walking_distance"": 65
    }
  ]
}

I'll put the result into a simple JSON layout so it's easy to paste or check. It should follow this shape:

{
  ""solution"": {
    ""selected"": [<seat_to_open>, <seat_to_open>, ...],
    ""assignments"": [<chosen_open_seat>, <chosen_open_seat>, ...]
  }
}

""selected"" is where I'll list which seating spots we decide to open, and ""assignments"" is a parallel list (one entry per street segment in the same order as the input) saying which opened seat that segment is assigned to. Think of it like filling out a form: pick the open seats, then for each street segment write the chosen seat next to it.

This JSON is just a sketch of the expected shape, not the actual answer yet. All identifiers in the final filled-in JSON must match the instance input exactly — no renaming and no new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 25, 29, 41, 49, 49, 39], [25, 0, 36, 24, 54, 52, 38], [29, 36, 0, 27, 58, 59, 52], [41, 24, 27, 0, 53, 70, 49], [49, 54, 58, 53, 0, 77, 61], [49, 52, 59, 70, 77, 0, 65], [39, 38, 52, 49, 61, 65, 0]], 'p': 1, 'objective': 49.0}","{'facilities': [0], 'assignments': [0, 0, 0, 0, 0, 0, 0]}",49.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 25, 'C': 29, 'D': 41, 'E': 49, 'F': 49, 'G': 39}}, {'id': 'B', 'distances': {'A': 25, 'B': 0, 'C': 36, 'D': 24, 'E': 54, 'F': 52, 'G': 38}}, {'id': 'C', 'distances': {'A': 29, 'B': 36, 'C': 0, 'D': 27, 'E': 58, 'F': 59, 'G': 52}}, {'id': 'D', 'distances': {'A': 41, 'B': 24, 'C': 27, 'D': 0, 'E': 53, 'F': 70, 'G': 49}}, {'id': 'E', 'distances': {'A': 49, 'B': 54, 'C': 58, 'D': 53, 'E': 0, 'F': 77, 'G': 61}}, {'id': 'F', 'distances': {'A': 49, 'B': 52, 'C': 59, 'D': 70, 'E': 77, 'F': 0, 'G': 65}}, {'id': 'G', 'distances': {'A': 39, 'B': 38, 'C': 52, 'D': 49, 'E': 61, 'F': 65, 'G': 0}}], 'objective': 49.0}","{'selected': ['A'], 'assignments': ['A', 'A', 'A', 'A', 'A', 'A', 'A']}",23,json,names
PCENTER,PCENTER,"I was picturing a telecom planner with a checklist: pick a fixed number of equipment sites from a list of possible spots, then hook every neighborhood up to exactly one of the active sites so nothing is left out or doubled. The success of a plan is judged by the neighborhood that ends up the farthest away — after all connections are made, find the single largest link distance from any neighborhood to its assigned site, and the smaller that “worst” distance is, the better the choice of sites. All neighborhoods must be connected and each to only one site. Concrete details for the specific instance are shown below.

# num_candidate_sites=5
# num_sites_to_activate=1
# location_ids=A, B, C, D, E
from_location_id,to_location_id,link_distance
A,B,123
A,C,177
A,D,206
A,E,117
B,A,123
B,C,115
B,D,135
B,E,187
C,A,177
C,B,115
C,D,78
C,E,172
D,A,206
D,B,135
D,C,78
D,E,155
E,A,117
E,B,187
E,C,172
E,D,155

If you want to hand me the results in a tidy form, here's the little JSON sketch I expect — just a simple container with which sites you decided to open and which open site each neighborhood gets paired with.

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_site>, <assigned_site>, ...]
  }
}

Think of ""selected"" as the checklist of sites you flipped on, and ""assignments"" as the lineup that tells me, for each neighborhood in the original order, which of the open sites it's hooked up to. Super informal: ""selected"" = open spots, ""assignments"" = who goes to which open spot.

This JSON is just a shape example — not the actual plan. One important note: use the exact identifiers from the instance input when you fill this in — do not rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 123, 177, 206, 117], [123, 0, 115, 135, 187], [177, 115, 0, 78, 172], [206, 135, 78, 0, 155], [117, 187, 172, 155, 0]], 'p': 1, 'objective': 177.0}","{'facilities': [2], 'assignments': [2, 2, 2, 2, 2]}",177.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 123, 'C': 177, 'D': 206, 'E': 117}}, {'id': 'B', 'distances': {'A': 123, 'B': 0, 'C': 115, 'D': 135, 'E': 187}}, {'id': 'C', 'distances': {'A': 177, 'B': 115, 'C': 0, 'D': 78, 'E': 172}}, {'id': 'D', 'distances': {'A': 206, 'B': 135, 'C': 78, 'D': 0, 'E': 155}}, {'id': 'E', 'distances': {'A': 117, 'B': 187, 'C': 172, 'D': 155, 'E': 0}}], 'objective': 177.0}","{'selected': ['C'], 'assignments': ['C', 'C', 'C', 'C', 'C']}",24,csv,names
PCENTER,PCENTER,"We were planning where the mobile clinics should park for a week: there are several candidate spots, but only a set number can be opened on any day. The task is to choose which stops to open and then attach every village to one of those chosen stops — every village gets exactly one assignment, no duplicates or omissions. To judge a setup, check how far each village has to travel to its assigned stop, pick out the longest of those trips, and try to make that longest trip as short as possible. The concrete details will be shown below.

# total_locations_count=6
# clinics_to_open=1
# location_ids=A, B, C, D, E, F
from_location_id,to_location_id,travel_distance
A,B,35
A,C,54
A,D,50
A,E,46
A,F,34
B,A,35
B,C,49
B,D,53
B,E,46
B,F,42
C,A,54
C,B,49
C,D,38
C,E,51
C,F,30
D,A,50
D,B,53
D,C,38
D,E,56
D,F,24
E,A,46
E,B,46
E,C,51
E,D,56
E,F,44
F,A,34
F,B,42
F,C,30
F,D,24
F,E,44

Also, when you send the actual pick-and-assign, please follow this simple JSON shape so I can read it easily:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_open_site>, <assigned_open_site>, ...]
  }
}

""selected"" is where you list the stops you plan to open. ""assignments"" lists, for each village in the input order, which opened stop that village will use. Think of it like filling out a form: first the chosen stops, then one assignment per village. This is just the expected shape — not your final choices.

Quick reminder: use the exact identifiers from the instance input — do not rename them or invent new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 35, 54, 50, 46, 34], [35, 0, 49, 53, 46, 42], [54, 49, 0, 38, 51, 30], [50, 53, 38, 0, 56, 24], [46, 46, 51, 56, 0, 44], [34, 42, 30, 24, 44, 0]], 'p': 1, 'objective': 44.0}","{'facilities': [5], 'assignments': [5, 5, 5, 5, 5, 5]}",44.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 35, 'C': 54, 'D': 50, 'E': 46, 'F': 34}}, {'id': 'B', 'distances': {'A': 35, 'B': 0, 'C': 49, 'D': 53, 'E': 46, 'F': 42}}, {'id': 'C', 'distances': {'A': 54, 'B': 49, 'C': 0, 'D': 38, 'E': 51, 'F': 30}}, {'id': 'D', 'distances': {'A': 50, 'B': 53, 'C': 38, 'D': 0, 'E': 56, 'F': 24}}, {'id': 'E', 'distances': {'A': 46, 'B': 46, 'C': 51, 'D': 56, 'E': 0, 'F': 44}}, {'id': 'F', 'distances': {'A': 34, 'B': 42, 'C': 30, 'D': 24, 'E': 44, 'F': 0}}], 'objective': 44.0}","{'selected': ['F'], 'assignments': ['F', 'F', 'F', 'F', 'F', 'F']}",25,csv,names
PCENTER,PCENTER,"Many people drive this campus, so the facilities team needs to pick a fixed number of chargers from approved spots and assign each building to one of them. The way they’ll compare different setups is to take every building’s distance to its assigned charger and then look at the biggest of those distances — a better setup is the one with the smaller biggest distance. Every building must be paired with exactly one charger, and the actual pad locations and distance numbers are provided below.

# total_candidate_locations=6
# chargers_to_open=1
# campus_location_ids=0, 1, 2, 3, 4, 5
origin_location_id,destination_location_id,travel_distance_between_locations
0,1,45
0,2,145
0,3,172
0,4,140
0,5,253
1,0,45
1,2,147
1,3,147
1,4,127
1,5,208
2,0,145
2,1,147
2,3,139
2,4,142
2,5,269
3,0,172
3,1,147
3,2,139
3,4,98
3,5,207
4,0,140
4,1,127
4,2,142
4,3,98
4,5,180
5,0,253
5,1,208
5,2,269
5,3,207
5,4,180

When you send back the plan, please use this little JSON layout so it's easy to read and parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

""selected"" is just the list of charger pads you want to open, and ""assignments"" lists, for every building in the same order as the instance, which opened pad it's paired with. Think of it like filling out a short form — which pads are open, and which pad each building uses.

This is just a sketch of the shape I need, not the actual answer.

Please use the identifiers exactly as they appear in the instance input — do not rename them or create new labels.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 45, 145, 172, 140, 253], [45, 0, 147, 147, 127, 208], [145, 147, 0, 139, 142, 269], [172, 147, 139, 0, 98, 207], [140, 127, 142, 98, 0, 180], [253, 208, 269, 207, 180, 0]], 'p': 1, 'objective': 180.0}","{'facilities': [4], 'assignments': [4, 4, 4, 4, 4, 4]}",180.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 45, 2: 145, 3: 172, 4: 140, 5: 253}}, {'id': 1, 'distances': {0: 45, 1: 0, 2: 147, 3: 147, 4: 127, 5: 208}}, {'id': 2, 'distances': {0: 145, 1: 147, 2: 0, 3: 139, 4: 142, 5: 269}}, {'id': 3, 'distances': {0: 172, 1: 147, 2: 139, 3: 0, 4: 98, 5: 207}}, {'id': 4, 'distances': {0: 140, 1: 127, 2: 142, 3: 98, 4: 0, 5: 180}}, {'id': 5, 'distances': {0: 253, 1: 208, 2: 269, 3: 207, 4: 180, 5: 0}}], 'objective': 180.0}","{'selected': [4], 'assignments': [4, 4, 4, 4, 4, 4]}",26,csv,0
PCENTER,PCENTER,"We imagine the park team picking a set number of first aid stations out of the available clearings and then assigning every trailhead to one of the stations that stay open. The goal is simple in plain terms: look at the distance from every trailhead to the post it’s assigned to, find the one that’s farthest away, and make that farthest distance as small as possible. Every trailhead must be attached to one and only one open post, and only the selected clearings become posts. The concrete details follow below.

We list 8 candidate locations, will open 1 first aid posts, and identify each location by these IDs: 1, 2, 3, 4, 5, 6, 7, 8.
We record the trail distance between 1 and 2 as 53.
We record the trail distance between 1 and 3 as 88.
We record the trail distance between 1 and 4 as 64.
We record the trail distance between 1 and 5 as 41.
We record the trail distance between 1 and 6 as 71.
We record the trail distance between 1 and 7 as 51.
We record the trail distance between 1 and 8 as 55.
We record the trail distance between 2 and 1 as 53.
We record the trail distance between 2 and 3 as 71.
We record the trail distance between 2 and 4 as 39.
We record the trail distance between 2 and 5 as 40.
We record the trail distance between 2 and 6 as 26.
We record the trail distance between 2 and 7 as 38.
We record the trail distance between 2 and 8 as 52.
We record the trail distance between 3 and 1 as 88.
We record the trail distance between 3 and 2 as 71.
We record the trail distance between 3 and 4 as 54.
We record the trail distance between 3 and 5 as 55.
We record the trail distance between 3 and 6 as 57.
We record the trail distance between 3 and 7 as 37.
We record the trail distance between 3 and 8 as 49.
We record the trail distance between 4 and 1 as 64.
We record the trail distance between 4 and 2 as 39.
We record the trail distance between 4 and 3 as 54.
We record the trail distance between 4 and 5 as 23.
We record the trail distance between 4 and 6 as 57.
We record the trail distance between 4 and 7 as 46.
We record the trail distance between 4 and 8 as 37.
We record the trail distance between 5 and 1 as 41.
We record the trail distance between 5 and 2 as 40.
We record the trail distance between 5 and 3 as 55.
We record the trail distance between 5 and 4 as 23.
We record the trail distance between 5 and 6 as 39.
We record the trail distance between 5 and 7 as 26.
We record the trail distance between 5 and 8 as 14.
We record the trail distance between 6 and 1 as 71.
We record the trail distance between 6 and 2 as 26.
We record the trail distance between 6 and 3 as 57.
We record the trail distance between 6 and 4 as 57.
We record the trail distance between 6 and 5 as 39.
We record the trail distance between 6 and 7 as 50.
We record the trail distance between 6 and 8 as 42.
We record the trail distance between 7 and 1 as 51.
We record the trail distance between 7 and 2 as 38.
We record the trail distance between 7 and 3 as 37.
We record the trail distance between 7 and 4 as 46.
We record the trail distance between 7 and 5 as 26.
We record the trail distance between 7 and 6 as 50.
We record the trail distance between 7 and 8 as 32.
We record the trail distance between 8 and 1 as 55.
We record the trail distance between 8 and 2 as 52.
We record the trail distance between 8 and 3 as 49.
We record the trail distance between 8 and 4 as 37.
We record the trail distance between 8 and 5 as 14.
We record the trail distance between 8 and 6 as 42.
We record the trail distance between 8 and 7 as 32.
We will use these distances to assign every trailhead to an open post and minimize the maximum assigned distance.

You can just return the solution in a tiny JSON snippet like this — nothing fancy, just fill the arrays with the clearing IDs and assignments.

{
  ""solution"": {
    ""selected"": [<clearing_to_open>, <clearing_to_open>, ...],
    ""assignments"": [<assigned_open_clearing>, <assigned_open_clearing>, ...]
  }
}

""selected"" is the list of clearings you decide to open as first-aid posts. ""assignments"" says, for each trailhead in the instance (in the same order the instance lists them), which open clearing that trailhead is assigned to. Think of it like a simple form: pick the open posts, then for each trailhead name the post it uses.

This JSON is just the shape I expect you to follow — it's a sketch, not the filled-in answer.

Please be careful: use the exact identifiers from the instance input, with no renaming and no new labels. Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'distance_matrix': [[0, 53, 88, 64, 41, 71, 51, 55], [53, 0, 71, 39, 40, 26, 38, 52], [88, 71, 0, 54, 55, 57, 37, 49], [64, 39, 54, 0, 23, 57, 46, 37], [41, 40, 55, 23, 0, 39, 26, 14], [71, 26, 57, 57, 39, 0, 50, 42], [51, 38, 37, 46, 26, 50, 0, 32], [55, 52, 49, 37, 14, 42, 32, 0]], 'p': 1, 'objective': 51.0}","{'facilities': [6], 'assignments': [6, 6, 6, 6, 6, 6, 6, 6]}",51.0,"{'problem_type': 'PCENTER', 'num_nodes': 8, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 53, 3: 88, 4: 64, 5: 41, 6: 71, 7: 51, 8: 55}}, {'id': 2, 'distances': {1: 53, 2: 0, 3: 71, 4: 39, 5: 40, 6: 26, 7: 38, 8: 52}}, {'id': 3, 'distances': {1: 88, 2: 71, 3: 0, 4: 54, 5: 55, 6: 57, 7: 37, 8: 49}}, {'id': 4, 'distances': {1: 64, 2: 39, 3: 54, 4: 0, 5: 23, 6: 57, 7: 46, 8: 37}}, {'id': 5, 'distances': {1: 41, 2: 40, 3: 55, 4: 23, 5: 0, 6: 39, 7: 26, 8: 14}}, {'id': 6, 'distances': {1: 71, 2: 26, 3: 57, 4: 57, 5: 39, 6: 0, 7: 50, 8: 42}}, {'id': 7, 'distances': {1: 51, 2: 38, 3: 37, 4: 46, 5: 26, 6: 50, 7: 0, 8: 32}}, {'id': 8, 'distances': {1: 55, 2: 52, 3: 49, 4: 37, 5: 14, 6: 42, 7: 32, 8: 0}}], 'objective': 51.0}","{'selected': [7], 'assignments': [7, 7, 7, 7, 7, 7, 7, 7]}",27,nl,1
PCENTER,PCENTER,"We operate a network of shops and need to decide which few storefronts will host small fulfillment kiosks. The plan is to pick the required number of kiosks, then assign every sales area to exactly one open kiosk so every area is covered and none are assigned twice. To compare different choices, we look at the farthest any area has to travel to its kiosk — the smaller that farthest trip, the better the setup. The specific sites and the distances between them are shown below.

There are 5 candidate storefronts; we must open 1 kiosks, and the location IDs are 1, 2, 3, 4, 5.
We measure the dispatch distance from 1 to 2 as 66.
We measure the dispatch distance from 1 to 3 as 52.
We measure the dispatch distance from 1 to 4 as 39.
We measure the dispatch distance from 1 to 5 as 39.
We measure the dispatch distance from 2 to 1 as 66.
We measure the dispatch distance from 2 to 3 as 68.
We measure the dispatch distance from 2 to 4 as 41.
We measure the dispatch distance from 2 to 5 as 37.
We measure the dispatch distance from 3 to 1 as 52.
We measure the dispatch distance from 3 to 2 as 68.
We measure the dispatch distance from 3 to 4 as 54.
We measure the dispatch distance from 3 to 5 as 74.
We measure the dispatch distance from 4 to 1 as 39.
We measure the dispatch distance from 4 to 2 as 41.
We measure the dispatch distance from 4 to 3 as 54.
We measure the dispatch distance from 4 to 5 as 29.
We measure the dispatch distance from 5 to 1 as 39.
We measure the dispatch distance from 5 to 2 as 37.
We measure the dispatch distance from 5 to 3 as 74.
We measure the dispatch distance from 5 to 4 as 29.
We will use these distances to evaluate the worst-case assigned dispatch distance for the chosen 1 kiosks.

Also, to keep things tidy, please send your choice in this little JSON shape — nothing fancy, just the skeleton below that says which sites you open and which open site each area is assigned to.

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

This is just a sketch of the shape I expect: ""selected"" is the list of sites where we put kiosks, and ""assignments"" lists, for every sales area in the same order as the instance, which open site it's assigned to. Light and informal — think of it like filling in two columns on a form, not a full report.

One more thing: use the exact identifiers from the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 66, 52, 39, 39], [66, 0, 68, 41, 37], [52, 68, 0, 54, 74], [39, 41, 54, 0, 29], [39, 37, 74, 29, 0]], 'p': 1, 'objective': 54.0}","{'facilities': [3], 'assignments': [3, 3, 3, 3, 3]}",54.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 66, 3: 52, 4: 39, 5: 39}}, {'id': 2, 'distances': {1: 66, 2: 0, 3: 68, 4: 41, 5: 37}}, {'id': 3, 'distances': {1: 52, 2: 68, 3: 0, 4: 54, 5: 74}}, {'id': 4, 'distances': {1: 39, 2: 41, 3: 54, 4: 0, 5: 29}}, {'id': 5, 'distances': {1: 39, 2: 37, 3: 74, 4: 29, 5: 0}}], 'objective': 54.0}","{'selected': [4], 'assignments': [4, 4, 4, 4, 4]}",28,markdown_table,1
PCENTER,PCENTER,"We’re picturing the event team laying out a few information desks around the galleries: they choose a set number of booth locations from the proposed spots, and then tie each exhibit area to one of those booths so every area is covered. The rule is the number of booths is fixed and every exhibit zone must be linked to exactly one booth — nothing can be left out or assigned twice. To judge any layout, check each zone’s walking distance to its linked booth, find the biggest distance among them, and aim to shrink that biggest walk as much as possible. The actual list of spots and distance numbers appears below.

Below are the 5 candidate spots and their pairwise walking distances; open exactly 1 booths and assign each of the 0, 1, 2, 3, 4 locations to one opened booth.
We record the walking distance from 0 to 1 as 40.
We record the walking distance from 0 to 2 as 39.
We record the walking distance from 0 to 3 as 37.
We record the walking distance from 0 to 4 as 29.
We record the walking distance from 1 to 0 as 40.
We record the walking distance from 1 to 2 as 32.
We record the walking distance from 1 to 3 as 48.
We record the walking distance from 1 to 4 as 37.
We record the walking distance from 2 to 0 as 39.
We record the walking distance from 2 to 1 as 32.
We record the walking distance from 2 to 3 as 46.
We record the walking distance from 2 to 4 as 40.
We record the walking distance from 3 to 0 as 37.
We record the walking distance from 3 to 1 as 48.
We record the walking distance from 3 to 2 as 46.
We record the walking distance from 3 to 4 as 48.
We record the walking distance from 4 to 0 as 29.
We record the walking distance from 4 to 1 as 37.
We record the walking distance from 4 to 2 as 40.
We record the walking distance from 4 to 3 as 48.
We’ll use these numbers to evaluate assignments and minimize the maximum walk.

Oh, and when you send the layout back, please use this little JSON sketch so it's easy to parse. It should follow this shape exactly:

{
  ""solution"": {
    ""selected"": [<booth_to_open>, <booth_to_open>, ...],
    ""assignments"": [<chosen_open_booth>, <chosen_open_booth>, ...]
  }
}

""selected"" is the list of booth locations you decide to open. ""assignments"" is a list at the same length as the list of exhibit areas in the instance: each entry names which opened booth that area is linked to. Think of it like filling out a simple form — which booths are open, and which booth covers each area.

This JSON is just a template showing the expected shape, not the real answer. Please use the exact identifiers from the instance input for any locations or areas — do not rename them or invent new labels. 

Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 40, 39, 37, 29], [40, 0, 32, 48, 37], [39, 32, 0, 46, 40], [37, 48, 46, 0, 48], [29, 37, 40, 48, 0]], 'p': 1, 'objective': 40.0}","{'facilities': [0], 'assignments': [0, 0, 0, 0, 0]}",40.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 40, 2: 39, 3: 37, 4: 29}}, {'id': 1, 'distances': {0: 40, 1: 0, 2: 32, 3: 48, 4: 37}}, {'id': 2, 'distances': {0: 39, 1: 32, 2: 0, 3: 46, 4: 40}}, {'id': 3, 'distances': {0: 37, 1: 48, 2: 46, 3: 0, 4: 48}}, {'id': 4, 'distances': {0: 29, 1: 37, 2: 40, 3: 48, 4: 0}}], 'objective': 40.0}","{'selected': [0], 'assignments': [0, 0, 0, 0, 0]}",29,nl,0
PCENTER,PCENTER,"As part of the event planning, the coordinator must open a fixed number of concession bases from a set of candidate locations and assign each crowd cluster to one of the opened bases. Every cluster has to be assigned to exactly one base and only the chosen number of bases may be used. To compare choices, look at each cluster’s walk to its assigned base, take the longest of those walks, and favor the plan that minimizes that longest walk. The specific locations and distances are shown below.

This instance contains 6 total locations; the coordinator must open exactly 2 concession bases and may choose from these location identifiers: 0, 1, 2, 3, 4, 5.
From location 0 to location 1, the walking distance is 44.
From location 0 to location 2, the walking distance is 35.
From location 0 to location 3, the walking distance is 35.
From location 0 to location 4, the walking distance is 32.
From location 0 to location 5, the walking distance is 39.
From location 1 to location 0, the walking distance is 44.
From location 1 to location 2, the walking distance is 42.
From location 1 to location 3, the walking distance is 34.
From location 1 to location 4, the walking distance is 38.
From location 1 to location 5, the walking distance is 43.
From location 2 to location 0, the walking distance is 35.
From location 2 to location 1, the walking distance is 42.
From location 2 to location 3, the walking distance is 29.
From location 2 to location 4, the walking distance is 33.
From location 2 to location 5, the walking distance is 36.
From location 3 to location 0, the walking distance is 35.
From location 3 to location 1, the walking distance is 34.
From location 3 to location 2, the walking distance is 29.
From location 3 to location 4, the walking distance is 29.
From location 3 to location 5, the walking distance is 32.
From location 4 to location 0, the walking distance is 32.
From location 4 to location 1, the walking distance is 38.
From location 4 to location 2, the walking distance is 33.
From location 4 to location 3, the walking distance is 29.
From location 4 to location 5, the walking distance is 27.
From location 5 to location 0, the walking distance is 39.
From location 5 to location 1, the walking distance is 43.
From location 5 to location 2, the walking distance is 36.
From location 5 to location 3, the walking distance is 32.
From location 5 to location 4, the walking distance is 27.
The coordinator will use these distances to assess assignment plans under the requirement to open 2 bases.

If you want to hand in a proposed plan, a relaxed way to do it is to follow this little JSON sketch for the shape of the answer:

{
  ""solution"": {
    ""selected"": [""<site_to_open>"", ""<site_to_open>"", ...],
    ""assignments"": [""<chosen_site>"", ""<chosen_site>"", ...]
  }
}

This just means: ""selected"" is the list of concession sites you decide to open, and ""assignments"" lists, for each crowd cluster in the instance order, which opened site that cluster will use. Think of it like filling out a simple form — the JSON above is only the shape I need, not the actual plan.

Please be sure to use the exact identifiers as they appear in the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 44, 35, 35, 32, 39], [44, 0, 42, 34, 38, 43], [35, 42, 0, 29, 33, 36], [35, 34, 29, 0, 29, 32], [32, 38, 33, 29, 0, 27], [39, 43, 36, 32, 27, 0]], 'p': 2, 'objective': 33.0}","{'facilities': [1, 4], 'assignments': [4, 1, 4, 4, 4, 4]}",33.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 2, 'sites': [{'id': 0, 'distances': {0: 0, 1: 44, 2: 35, 3: 35, 4: 32, 5: 39}}, {'id': 1, 'distances': {0: 44, 1: 0, 2: 42, 3: 34, 4: 38, 5: 43}}, {'id': 2, 'distances': {0: 35, 1: 42, 2: 0, 3: 29, 4: 33, 5: 36}}, {'id': 3, 'distances': {0: 35, 1: 34, 2: 29, 3: 0, 4: 29, 5: 32}}, {'id': 4, 'distances': {0: 32, 1: 38, 2: 33, 3: 29, 4: 0, 5: 27}}, {'id': 5, 'distances': {0: 39, 1: 43, 2: 36, 3: 32, 4: 27, 5: 0}}], 'objective': 33.0}","{'selected': [1, 4], 'assignments': [4, 1, 4, 4, 4, 4]}",30,nl,0
PCENTER,PCENTER,"We’ve got a pile of potential sites and a clear limit on how many rapid-response stations we can open, so the job is to pick which ones to use. Each district must go to one and only one open station, and what makes a plan better is how close the most remote district ends up being — measure every district’s ride to its station, find the longest ride, and aim to make that longest ride as short as possible. The exact locations and the distance numbers are given below.

Here are the 5 candidate locations; we must open exactly 1 bases, and the location identifiers are 1, 2, 3, 4, 5.
From location 1 to location 2: 56.
From location 1 to location 3: 17.
From location 1 to location 4: 45.
From location 1 to location 5: 52.
From location 2 to location 1: 56.
From location 2 to location 3: 45.
From location 2 to location 4: 54.
From location 2 to location 5: 56.
From location 3 to location 1: 17.
From location 3 to location 2: 45.
From location 3 to location 4: 34.
From location 3 to location 5: 38.
From location 4 to location 1: 45.
From location 4 to location 2: 54.
From location 4 to location 3: 34.
From location 4 to location 5: 42.
From location 5 to location 1: 52.
From location 5 to location 2: 56.
From location 5 to location 3: 38.
From location 5 to location 4: 42.
Use these distances to assign every district and judge plans by the longest ride.

I'll sketch the shape of the reply so it's clear how to fill it in — a simple JSON object with the parts we need.

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_site>, <assigned_site>, ...]
  }
}

""selected"" is the list of sites you'll open. ""assignments"" gives, for each district in the same order as the instance, which opened site that district is assigned to. Think of it like a form: one list of chosen station locations, and one list saying which station each district uses.

This is just a template of the expected shape, not the actual answer — fill in the real identifiers from the instance.

Please be sure to use the exact identifiers from the input, with no renaming or new labels.

Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 56, 17, 45, 52], [56, 0, 45, 54, 56], [17, 45, 0, 34, 38], [45, 54, 34, 0, 42], [52, 56, 38, 42, 0]], 'p': 1, 'objective': 45.0}","{'facilities': [2], 'assignments': [2, 2, 2, 2, 2]}",45.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 56, 3: 17, 4: 45, 5: 52}}, {'id': 2, 'distances': {1: 56, 2: 0, 3: 45, 4: 54, 5: 56}}, {'id': 3, 'distances': {1: 17, 2: 45, 3: 0, 4: 34, 5: 38}}, {'id': 4, 'distances': {1: 45, 2: 54, 3: 34, 4: 0, 5: 42}}, {'id': 5, 'distances': {1: 52, 2: 56, 3: 38, 4: 42, 5: 0}}], 'objective': 45.0}","{'selected': [3], 'assignments': [3, 3, 3, 3, 3]}",31,markdown_table,1
PCENTER,PCENTER,"There’s a small practical challenge on the table: a fixed number of transformers must be installed on chosen candidate plots, and each hamlet must be tied to exactly one of the installed transformers — no hamlet can be skipped or linked to more than one. A plan’s quality is decided by the single largest travel distance any hamlet has to its assigned transformer: compute each hamlet’s distance to its transformer and take the maximum — the lower that maximum, the better. The specific locations and distance data appear below.

# total_candidate_locations=5
# num_transformers_to_install=1
# location_ids=1, 2, 3, 4, 5
origin_location_id,destination_location_id,travel_distance
1,2,23
1,3,39
1,4,27
1,5,37
2,1,23
2,3,39
2,4,18
2,5,33
3,1,39
3,2,39
3,4,35
3,5,48
4,1,27
4,2,18
4,3,35
4,5,34
5,1,37
5,2,33
5,3,48
5,4,34

Also, when you send the plan back, please stick to this simple JSON layout so it's easy to read and check. Here's the shape I expect:

{
  ""solution"": {
    ""selected"": [<plot_to_open>, <plot_to_open>, ...],
    ""assignments"": [<assigned_open_plot>, <assigned_open_plot>, ...]
  }
}

""selected"" is just the list of plots where you'll install transformers. ""assignments"" is a list of the chosen-open-plot for each hamlet (in the same order the instance lists the hamlets). Think of it like filling out a short form: which plots we open, and for each hamlet which opened plot it uses.

This JSON is only a sketch of the expected shape — not the actual answer. Also, be careful: use the identifiers exactly as they appear in the instance input — do not rename them or invent new labels. 

- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 23, 39, 27, 37], [23, 0, 39, 18, 33], [39, 39, 0, 35, 48], [27, 18, 35, 0, 34], [37, 33, 48, 34, 0]], 'p': 1, 'objective': 35.0}","{'facilities': [3], 'assignments': [3, 3, 3, 3, 3]}",35.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 23, 3: 39, 4: 27, 5: 37}}, {'id': 2, 'distances': {1: 23, 2: 0, 3: 39, 4: 18, 5: 33}}, {'id': 3, 'distances': {1: 39, 2: 39, 3: 0, 4: 35, 5: 48}}, {'id': 4, 'distances': {1: 27, 2: 18, 3: 35, 4: 0, 5: 34}}, {'id': 5, 'distances': {1: 37, 2: 33, 3: 48, 4: 34, 5: 0}}], 'objective': 35.0}","{'selected': [4], 'assignments': [4, 4, 4, 4, 4]}",32,csv,1
PCENTER,PCENTER,"I heard the mobile library is trying to pick a fixed number of pickup spots from a list of possible locations, and then link every neighborhood to exactly one of those chosen spots so nobody’s left out or assigned twice. The trick is to pick the stops so that the person who has to travel the farthest ends up with the shortest possible trip — you figure that by looking at each neighborhood’s distance to its assigned stop and taking the largest of those distances. The exact locations and travel distances will be shown below.

Below are the 6 candidate locations; open exactly 2 pickup spots. Their identifiers are 1, 2, 3, 4, 5, 6.
From neighborhood 1 to candidate stop 2, the travel distance is 47.
From neighborhood 1 to candidate stop 3, the travel distance is 39.
From neighborhood 1 to candidate stop 4, the travel distance is 40.
From neighborhood 1 to candidate stop 5, the travel distance is 15.
From neighborhood 1 to candidate stop 6, the travel distance is 44.
From neighborhood 2 to candidate stop 1, the travel distance is 47.
From neighborhood 2 to candidate stop 3, the travel distance is 56.
From neighborhood 2 to candidate stop 4, the travel distance is 41.
From neighborhood 2 to candidate stop 5, the travel distance is 51.
From neighborhood 2 to candidate stop 6, the travel distance is 63.
From neighborhood 3 to candidate stop 1, the travel distance is 39.
From neighborhood 3 to candidate stop 2, the travel distance is 56.
From neighborhood 3 to candidate stop 4, the travel distance is 38.
From neighborhood 3 to candidate stop 5, the travel distance is 26.
From neighborhood 3 to candidate stop 6, the travel distance is 62.
From neighborhood 4 to candidate stop 1, the travel distance is 40.
From neighborhood 4 to candidate stop 2, the travel distance is 41.
From neighborhood 4 to candidate stop 3, the travel distance is 38.
From neighborhood 4 to candidate stop 5, the travel distance is 37.
From neighborhood 4 to candidate stop 6, the travel distance is 40.
From neighborhood 5 to candidate stop 1, the travel distance is 15.
From neighborhood 5 to candidate stop 2, the travel distance is 51.
From neighborhood 5 to candidate stop 3, the travel distance is 26.
From neighborhood 5 to candidate stop 4, the travel distance is 37.
From neighborhood 5 to candidate stop 6, the travel distance is 58.
From neighborhood 6 to candidate stop 1, the travel distance is 44.
From neighborhood 6 to candidate stop 2, the travel distance is 63.
From neighborhood 6 to candidate stop 3, the travel distance is 62.
From neighborhood 6 to candidate stop 4, the travel distance is 40.
From neighborhood 6 to candidate stop 5, the travel distance is 58.
Use these travel distances to assign every neighborhood so you can minimize the maximum trip any patron must take.

Also, when you're ready to send the answer, a handy, informal way to show it is with a small JSON snippet like this:

{
  ""solution"": {
    ""selected"": [<pickup_spot>, <pickup_spot>, ...],
    ""assignments"": [<chosen_pickup_spot>, <chosen_pickup_spot>, ...]
  }
}

Think of ""selected"" as the list of pickup spots the mobile library will open, and ""assignments"" as, for each neighborhood in the instance, which of the opened pickup spots that neighborhood is assigned to. It's a simple sketch of the shape we expect — not the actual answer.

Please make sure to use the exact identifiers from the instance input — do not rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 47, 39, 40, 15, 44], [47, 0, 56, 41, 51, 63], [39, 56, 0, 38, 26, 62], [40, 41, 38, 0, 37, 40], [15, 51, 26, 37, 0, 58], [44, 63, 62, 40, 58, 0]], 'p': 2, 'objective': 40.0}","{'facilities': [1, 3], 'assignments': [3, 1, 3, 3, 3, 3]}",40.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 2, 'sites': [{'id': 1, 'distances': {1: 0, 2: 47, 3: 39, 4: 40, 5: 15, 6: 44}}, {'id': 2, 'distances': {1: 47, 2: 0, 3: 56, 4: 41, 5: 51, 6: 63}}, {'id': 3, 'distances': {1: 39, 2: 56, 3: 0, 4: 38, 5: 26, 6: 62}}, {'id': 4, 'distances': {1: 40, 2: 41, 3: 38, 4: 0, 5: 37, 6: 40}}, {'id': 5, 'distances': {1: 15, 2: 51, 3: 26, 4: 37, 5: 0, 6: 58}}, {'id': 6, 'distances': {1: 44, 2: 63, 3: 62, 4: 40, 5: 58, 6: 0}}], 'objective': 40.0}","{'selected': [2, 4], 'assignments': [4, 2, 4, 4, 4, 4]}",33,nl,1
PCENTER,PCENTER,"At the last planning session the decision was made to open a specific number of tents at recommended spots and then funnel every residential area to exactly one open tent. The measure of a good plan is the longest trip any area has to make: after assigning everyone, take the maximum travel distance and try to minimize that number. Every area must be assigned once and only once, and the detailed list of sites and distances is shown below.

There are 7 locations in total; exactly 1 tents must be opened from the candidate locations 1, 2, 3, 4, 5, 6, 7.
From 1 to 2 the travel distance is 53.
From 1 to 3 the travel distance is 36.
From 1 to 4 the travel distance is 38.
From 1 to 5 the travel distance is 40.
From 1 to 6 the travel distance is 55.
From 1 to 7 the travel distance is 64.
From 2 to 1 the travel distance is 53.
From 2 to 3 the travel distance is 31.
From 2 to 4 the travel distance is 31.
From 2 to 5 the travel distance is 32.
From 2 to 6 the travel distance is 50.
From 2 to 7 the travel distance is 55.
From 3 to 1 the travel distance is 36.
From 3 to 2 the travel distance is 31.
From 3 to 4 the travel distance is 30.
From 3 to 5 the travel distance is 33.
From 3 to 6 the travel distance is 19.
From 3 to 7 the travel distance is 43.
From 4 to 1 the travel distance is 38.
From 4 to 2 the travel distance is 31.
From 4 to 3 the travel distance is 30.
From 4 to 5 the travel distance is 17.
From 4 to 6 the travel distance is 31.
From 4 to 7 the travel distance is 40.
From 5 to 1 the travel distance is 40.
From 5 to 2 the travel distance is 32.
From 5 to 3 the travel distance is 33.
From 5 to 4 the travel distance is 17.
From 5 to 6 the travel distance is 37.
From 5 to 7 the travel distance is 29.
From 6 to 1 the travel distance is 55.
From 6 to 2 the travel distance is 50.
From 6 to 3 the travel distance is 19.
From 6 to 4 the travel distance is 31.
From 6 to 5 the travel distance is 37.
From 6 to 7 the travel distance is 57.
From 7 to 1 the travel distance is 64.
From 7 to 2 the travel distance is 55.
From 7 to 3 the travel distance is 43.
From 7 to 4 the travel distance is 40.
From 7 to 5 the travel distance is 29.
From 7 to 6 the travel distance is 57.
Assign every area to one open tent so the maximum travel distance among assignments is minimized.

If you want to hand me the final plan, a simple JSON sketch like this is perfect — it just names which sites we open and who goes to which open site:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

Keep in mind what those pieces mean in plain terms: ""selected"" is the list of tent sites we decide to open, and ""assignments"" lists, for each residential area (in the same order as the instance), which opened site that area will go to. This is just the expected shape — not the actual answer.

Please also make sure to use the exact identifiers from the instance input — don't rename them or invent new labels. Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'distance_matrix': [[0, 53, 36, 38, 40, 55, 64], [53, 0, 31, 31, 32, 50, 55], [36, 31, 0, 30, 33, 19, 43], [38, 31, 30, 0, 17, 31, 40], [40, 32, 33, 17, 0, 37, 29], [55, 50, 19, 31, 37, 0, 57], [64, 55, 43, 40, 29, 57, 0]], 'p': 1, 'objective': 40.0}","{'facilities': [4], 'assignments': [4, 4, 4, 4, 4, 4, 4]}",40.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 53, 3: 36, 4: 38, 5: 40, 6: 55, 7: 64}}, {'id': 2, 'distances': {1: 53, 2: 0, 3: 31, 4: 31, 5: 32, 6: 50, 7: 55}}, {'id': 3, 'distances': {1: 36, 2: 31, 3: 0, 4: 30, 5: 33, 6: 19, 7: 43}}, {'id': 4, 'distances': {1: 38, 2: 31, 3: 30, 4: 0, 5: 17, 6: 31, 7: 40}}, {'id': 5, 'distances': {1: 40, 2: 32, 3: 33, 4: 17, 5: 0, 6: 37, 7: 29}}, {'id': 6, 'distances': {1: 55, 2: 50, 3: 19, 4: 31, 5: 37, 6: 0, 7: 57}}, {'id': 7, 'distances': {1: 64, 2: 55, 3: 43, 4: 40, 5: 29, 6: 57, 7: 0}}], 'objective': 40.0}","{'selected': [5], 'assignments': [5, 5, 5, 5, 5, 5, 5]}",34,markdown_table,1
PCENTER,PCENTER,"There’s a small chain deciding where to host pickup lockers: they can only activate a set number of storefronts from a list, and every customer must be routed to exactly one active pickup point. To judge any choice, imagine measuring how far each customer walks or drives to their assigned store, then spotting the single longest trip — the goal is to make that longest trip as short as it can be. The distances between addresses and storefronts are already known, and the full list of locations and numbers is shown below.

The instance lists 5 locations; 2 storefronts must be opened and the location identifiers are: A, B, C, D, E.
Distance from A to B is 62.
Distance from A to C is 29.
Distance from A to D is 68.
Distance from A to E is 100.
Distance from B to A is 62.
Distance from B to C is 47.
Distance from B to D is 34.
Distance from B to E is 89.
Distance from C to A is 29.
Distance from C to B is 47.
Distance from C to D is 48.
Distance from C to E is 77.
Distance from D to A is 68.
Distance from D to B is 34.
Distance from D to C is 48.
Distance from D to E is 80.
Distance from E to A is 100.
Distance from E to B is 89.
Distance from E to C is 77.
Distance from E to D is 80.
These entries let you assign customers to open pickup points and evaluate the maximum customer travel distance.

Quick note: when you reply with the actual choices, please follow this JSON layout so it's easy to parse.

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_open_site>, <assigned_open_site>, ...]
  }
}

""selected"" is where you list the storefronts you plan to activate (their IDs). ""assignments"" is a list, one entry per location in the instance, saying which open storefront that location is routed to. Think of the JSON as a simple form — it's just the shape I need, not the real answer.

Please use the exact identifiers from the instance input (no renaming, no new labels). 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 62, 29, 68, 100], [62, 0, 47, 34, 89], [29, 47, 0, 48, 77], [68, 34, 48, 0, 80], [100, 89, 77, 80, 0]], 'p': 2, 'objective': 48.0}","{'facilities': [2, 4], 'assignments': [2, 2, 2, 2, 4]}",48.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 2, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 62, 'C': 29, 'D': 68, 'E': 100}}, {'id': 'B', 'distances': {'A': 62, 'B': 0, 'C': 47, 'D': 34, 'E': 89}}, {'id': 'C', 'distances': {'A': 29, 'B': 47, 'C': 0, 'D': 48, 'E': 77}}, {'id': 'D', 'distances': {'A': 68, 'B': 34, 'C': 48, 'D': 0, 'E': 80}}, {'id': 'E', 'distances': {'A': 100, 'B': 89, 'C': 77, 'D': 80, 'E': 0}}], 'objective': 48.0}","{'selected': ['C', 'E'], 'assignments': ['C', 'C', 'C', 'C', 'E']}",35,markdown_table,names
PCENTER,PCENTER,"We’re trying to set up a limited number of antennas across town and attach every user location to one active antenna. The choice that wins is the one where the worst-off user — the one who’s farthest from their assigned antenna — has the shortest possible distance. Every location has to be assigned to a single turned-on antenna (no duplicates, no skipping), and only the allotted number of antennas can be used. The concrete locations and distances follow below.

{
  ""total_locations"": 6,
  ""antennas_to_activate"": 1,
  ""location_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F""
  ],
  ""data"": [
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 62
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 66
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 85
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 54
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 54
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 62
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 66
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 75
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 38
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 56
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 66
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 66
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 93
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 60
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 53
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 85
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 75
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 93
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 76
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 57
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 54
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 38
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 60
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 76
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 43
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 54
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 56
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 53
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 57
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 43
    }
  ]
}

Also, when you send the actual choices back, please use this simple JSON layout so it's easy to parse:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

Think of it like a little form: ""selected"" is the list of sites you decide to turn on, and ""assignments"" gives, for every location in the instance (in the same order they were listed), the site that location is attached to. Super casual — just fill those two lists and you're done.

This is just a sketch of the shape I need, not the final answer itself. One important rule: use the exact identifiers from the instance input — do not rename them or invent new labels. For example: ""Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".""","{'distance_matrix': [[0, 62, 66, 85, 54, 54], [62, 0, 66, 75, 38, 56], [66, 66, 0, 93, 60, 53], [85, 75, 93, 0, 76, 57], [54, 38, 60, 76, 0, 43], [54, 56, 53, 57, 43, 0]], 'p': 1, 'objective': 57.0}","{'facilities': [5], 'assignments': [5, 5, 5, 5, 5, 5]}",57.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 62, 'C': 66, 'D': 85, 'E': 54, 'F': 54}}, {'id': 'B', 'distances': {'A': 62, 'B': 0, 'C': 66, 'D': 75, 'E': 38, 'F': 56}}, {'id': 'C', 'distances': {'A': 66, 'B': 66, 'C': 0, 'D': 93, 'E': 60, 'F': 53}}, {'id': 'D', 'distances': {'A': 85, 'B': 75, 'C': 93, 'D': 0, 'E': 76, 'F': 57}}, {'id': 'E', 'distances': {'A': 54, 'B': 38, 'C': 60, 'D': 76, 'E': 0, 'F': 43}}, {'id': 'F', 'distances': {'A': 54, 'B': 56, 'C': 53, 'D': 57, 'E': 43, 'F': 0}}], 'objective': 57.0}","{'selected': ['F'], 'assignments': ['F', 'F', 'F', 'F', 'F', 'F']}",36,json,names
PCENTER,PCENTER,"Many people care about fair access, so imagine choosing a handful of service locations from a list of possible spots to cover a set of communities, with the distance between every pair already known. Each community must be assigned to one and only one chosen spot, and the best arrangement is the one that makes the longest required trip for any community as short as it can be. In practical terms, evaluate an arrangement by finding the community with the largest assigned travel distance — that largest distance is what gets reduced. The concrete data — locations and distances — are provided below.

The instance lists 6 distinct candidate locations, of which 2 must be opened; the ordered location identifiers are 0, 1, 2, 3, 4, 5.
The travel distance from 0 to 1 is 33.
The travel distance from 0 to 2 is 47.
The travel distance from 0 to 3 is 58.
The travel distance from 0 to 4 is 51.
The travel distance from 0 to 5 is 56.
The travel distance from 1 to 0 is 33.
The travel distance from 1 to 2 is 31.
The travel distance from 1 to 3 is 41.
The travel distance from 1 to 4 is 50.
The travel distance from 1 to 5 is 26.
The travel distance from 2 to 0 is 47.
The travel distance from 2 to 1 is 31.
The travel distance from 2 to 3 is 43.
The travel distance from 2 to 4 is 43.
The travel distance from 2 to 5 is 43.
The travel distance from 3 to 0 is 58.
The travel distance from 3 to 1 is 41.
The travel distance from 3 to 2 is 43.
The travel distance from 3 to 4 is 45.
The travel distance from 3 to 5 is 41.
The travel distance from 4 to 0 is 51.
The travel distance from 4 to 1 is 50.
The travel distance from 4 to 2 is 43.
The travel distance from 4 to 3 is 45.
The travel distance from 4 to 5 is 32.
The travel distance from 5 to 0 is 56.
The travel distance from 5 to 1 is 26.
The travel distance from 5 to 2 is 43.
The travel distance from 5 to 3 is 41.
The travel distance from 5 to 4 is 32.
Use these distance records to assess assignments and reduce the maximum assigned travel distance.

You can just reply using a simple JSON shape like this — nothing fancy, just the idea of which spots you open and who gets assigned where.

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

""selected"" is the list of sites you choose to open. ""assignments"" is, in the same order as the locations in the instance input, the open site each location is assigned to. Think of it like filling out a short form: pick the open sites, then for every community write which open site they're connected to.

This JSON is just the expected shape — a sketch, not the final filled-in answer. Please use the exact identifiers from the instance input; do not rename them or invent new labels. 

For example: ""Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".""","{'distance_matrix': [[0, 33, 47, 58, 51, 56], [33, 0, 31, 41, 50, 26], [47, 31, 0, 43, 43, 43], [58, 41, 43, 0, 45, 41], [51, 50, 43, 45, 0, 32], [56, 26, 43, 41, 32, 0]], 'p': 2, 'objective': 41.0}","{'facilities': [1, 4], 'assignments': [1, 1, 1, 1, 4, 4]}",41.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 2, 'sites': [{'id': 0, 'distances': {0: 0, 1: 33, 2: 47, 3: 58, 4: 51, 5: 56}}, {'id': 1, 'distances': {0: 33, 1: 0, 2: 31, 3: 41, 4: 50, 5: 26}}, {'id': 2, 'distances': {0: 47, 1: 31, 2: 0, 3: 43, 4: 43, 5: 43}}, {'id': 3, 'distances': {0: 58, 1: 41, 2: 43, 3: 0, 4: 45, 5: 41}}, {'id': 4, 'distances': {0: 51, 1: 50, 2: 43, 3: 45, 4: 0, 5: 32}}, {'id': 5, 'distances': {0: 56, 1: 26, 2: 43, 3: 41, 4: 32, 5: 0}}], 'objective': 41.0}","{'selected': [1, 4], 'assignments': [1, 1, 1, 1, 4, 4]}",37,nl,0
PCENTER,PCENTER,"Picture a neighborhood with a handful of candidate spots for chargers; the task is to choose a set number of those spots and attach every parking area to exactly one of the selected chargers, so nothing is left out or assigned twice. What counts as a good choice is how long the longest journey is — you compute each parking area’s distance to its assigned charger, take the biggest distance, and aim to make that biggest distance minimal. The specific locations and pairwise distances are listed below.

{
  ""total_locations"": 8,
  ""chargers_to_open"": 1,
  ""location_ids"": [
    ""A"",
    ""B"",
    ""C"",
    ""D"",
    ""E"",
    ""F"",
    ""G"",
    ""H""
  ],
  ""data"": [
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 53
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 54
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 47
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 27
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 47
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""G"",
      ""travel_distance"": 62
    },
    {
      ""origin_location_id"": ""A"",
      ""destination_location_id"": ""H"",
      ""travel_distance"": 12
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 53
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 58
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 59
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 45
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 10
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""G"",
      ""travel_distance"": 30
    },
    {
      ""origin_location_id"": ""B"",
      ""destination_location_id"": ""H"",
      ""travel_distance"": 50
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 54
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 58
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 33
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 56
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 55
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""G"",
      ""travel_distance"": 61
    },
    {
      ""origin_location_id"": ""C"",
      ""destination_location_id"": ""H"",
      ""travel_distance"": 49
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 47
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 59
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 33
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 51
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 49
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""G"",
      ""travel_distance"": 55
    },
    {
      ""origin_location_id"": ""D"",
      ""destination_location_id"": ""H"",
      ""travel_distance"": 35
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 27
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 45
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 56
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 51
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 40
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""G"",
      ""travel_distance"": 60
    },
    {
      ""origin_location_id"": ""E"",
      ""destination_location_id"": ""H"",
      ""travel_distance"": 38
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 47
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 10
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 55
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 49
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 40
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""G"",
      ""travel_distance"": 20
    },
    {
      ""origin_location_id"": ""F"",
      ""destination_location_id"": ""H"",
      ""travel_distance"": 50
    },
    {
      ""origin_location_id"": ""G"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 62
    },
    {
      ""origin_location_id"": ""G"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 30
    },
    {
      ""origin_location_id"": ""G"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 61
    },
    {
      ""origin_location_id"": ""G"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 55
    },
    {
      ""origin_location_id"": ""G"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 60
    },
    {
      ""origin_location_id"": ""G"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 20
    },
    {
      ""origin_location_id"": ""G"",
      ""destination_location_id"": ""H"",
      ""travel_distance"": 62
    },
    {
      ""origin_location_id"": ""H"",
      ""destination_location_id"": ""A"",
      ""travel_distance"": 12
    },
    {
      ""origin_location_id"": ""H"",
      ""destination_location_id"": ""B"",
      ""travel_distance"": 50
    },
    {
      ""origin_location_id"": ""H"",
      ""destination_location_id"": ""C"",
      ""travel_distance"": 49
    },
    {
      ""origin_location_id"": ""H"",
      ""destination_location_id"": ""D"",
      ""travel_distance"": 35
    },
    {
      ""origin_location_id"": ""H"",
      ""destination_location_id"": ""E"",
      ""travel_distance"": 38
    },
    {
      ""origin_location_id"": ""H"",
      ""destination_location_id"": ""F"",
      ""travel_distance"": 50
    },
    {
      ""origin_location_id"": ""H"",
      ""destination_location_id"": ""G"",
      ""travel_distance"": 62
    }
  ]
}

Also, when you give the final choice, please use this simple JSON layout so it's easy to parse and check — nothing fancy, just the shape below.

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

It's just a sketch of the shape I expect: ""selected"" lists the charger spots you decide to open, and ""assignments"" lists, for every parking area (in the same order as the instance), which opened spot it's assigned to. Keep it casual — think of ""selected"" as the boxes you tick and ""assignments"" as the little labels you stick on each parking area.

A quick reminder: use the exact identifiers from the instance input — don't rename them or invent new ones. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 53, 54, 47, 27, 47, 62, 12], [53, 0, 58, 59, 45, 10, 30, 50], [54, 58, 0, 33, 56, 55, 61, 49], [47, 59, 33, 0, 51, 49, 55, 35], [27, 45, 56, 51, 0, 40, 60, 38], [47, 10, 55, 49, 40, 0, 20, 50], [62, 30, 61, 55, 60, 20, 0, 62], [12, 50, 49, 35, 38, 50, 62, 0]], 'p': 1, 'objective': 55.0}","{'facilities': [5], 'assignments': [5, 5, 5, 5, 5, 5, 5, 5]}",55.0,"{'problem_type': 'PCENTER', 'num_nodes': 8, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 53, 'C': 54, 'D': 47, 'E': 27, 'F': 47, 'G': 62, 'H': 12}}, {'id': 'B', 'distances': {'A': 53, 'B': 0, 'C': 58, 'D': 59, 'E': 45, 'F': 10, 'G': 30, 'H': 50}}, {'id': 'C', 'distances': {'A': 54, 'B': 58, 'C': 0, 'D': 33, 'E': 56, 'F': 55, 'G': 61, 'H': 49}}, {'id': 'D', 'distances': {'A': 47, 'B': 59, 'C': 33, 'D': 0, 'E': 51, 'F': 49, 'G': 55, 'H': 35}}, {'id': 'E', 'distances': {'A': 27, 'B': 45, 'C': 56, 'D': 51, 'E': 0, 'F': 40, 'G': 60, 'H': 38}}, {'id': 'F', 'distances': {'A': 47, 'B': 10, 'C': 55, 'D': 49, 'E': 40, 'F': 0, 'G': 20, 'H': 50}}, {'id': 'G', 'distances': {'A': 62, 'B': 30, 'C': 61, 'D': 55, 'E': 60, 'F': 20, 'G': 0, 'H': 62}}, {'id': 'H', 'distances': {'A': 12, 'B': 50, 'C': 49, 'D': 35, 'E': 38, 'F': 50, 'G': 62, 'H': 0}}], 'objective': 55.0}","{'selected': ['F'], 'assignments': ['F', 'F', 'F', 'F', 'F', 'F', 'F', 'F']}",38,json,names
PCENTER,PCENTER,"On a sunny afternoon the team needs to decide where to open a fixed number of water refill stations and which picnic area will use which station. Every picnic area must be tied to one and only one open station — no duplicates and nothing left out. The way to tell if one setup is better than another is to look at the picnic area that walks the farthest to its station; the aim is to make that farthest walk as short as it can be. The exact candidate sites and distances will be shown below.

# num_locations=5
# kiosks_to_open=2
# location_ids=0, 1, 2, 3, 4
origin_location_id,destination_location_id,walking_distance
0,1,94
0,2,50
0,3,61
0,4,16
1,0,94
1,2,93
1,3,126
1,4,86
2,0,50
2,1,93
2,3,69
2,4,55
3,0,61
3,1,126
3,2,69
3,4,55
4,0,16
4,1,86
4,2,55
4,3,55

Also, to keep things machine-friendly, please give the final plan in this simple JSON shape so it's easy to check:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<assigned_site>, <assigned_site>, ...]
  }
}

This is just a sketch of the shape I need, not the actual answer. In plain terms: ""selected"" is the list of picnic sites you choose to open as refill stations, and ""assignments"" lists, for each picnic area in the problem input order, which open site that area will use. Keep it casual—think of filling in a short form.

All identifiers must be used exactly as they appear in the instance input — no renaming and no new labels.
- for example: ""Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".""","{'distance_matrix': [[0, 94, 50, 61, 16], [94, 0, 93, 126, 86], [50, 93, 0, 69, 55], [61, 126, 69, 0, 55], [16, 86, 55, 55, 0]], 'p': 2, 'objective': 55.0}","{'facilities': [1, 4], 'assignments': [4, 1, 4, 4, 4]}",55.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 2, 'sites': [{'id': 0, 'distances': {0: 0, 1: 94, 2: 50, 3: 61, 4: 16}}, {'id': 1, 'distances': {0: 94, 1: 0, 2: 93, 3: 126, 4: 86}}, {'id': 2, 'distances': {0: 50, 1: 93, 2: 0, 3: 69, 4: 55}}, {'id': 3, 'distances': {0: 61, 1: 126, 2: 69, 3: 0, 4: 55}}, {'id': 4, 'distances': {0: 16, 1: 86, 2: 55, 3: 55, 4: 0}}], 'objective': 55.0}","{'selected': [1, 4], 'assignments': [4, 1, 4, 4, 4]}",39,csv,0
PCENTER,PCENTER,"Picture this: a concert plaza with several candidate food stall locations and a handful of spots that can actually be opened. Every crowd group must be routed to exactly one of the open stalls — no splitting, no skipping. To see which routing is best, calculate how far each group would have to walk to its stall, pick the largest of those distances, and choose the plan where that largest distance is as small as possible. The specific data and map are shown below.

# total_candidate_spots=5
# stalls_to_open=1
# location_ids=A, B, C, D, E
from_location_id,to_location_id,walking_distance
A,B,23
A,C,39
A,D,34
A,E,25
B,A,23
B,C,47
B,D,38
B,E,36
C,A,39
C,B,47
C,D,29
C,E,48
D,A,34
D,B,38
D,C,29
D,E,29
E,A,25
E,B,36
E,C,48
E,D,29

Also, when you send the final plan, please follow this simple JSON layout so it’s easy to check automatically. Something like:

{
  ""solution"": {
    ""selected"": [<stall_to_open>, <stall_to_open>, ...],
    ""assignments"": [<chosen_open_stall>, <chosen_open_stall>, ...]
  }
}

Think of it as a little form:
- ""selected"" is the list of stalls you decide to open.
- ""assignments"" lists, for every location in the instance, which opened stall that location is assigned to.

This JSON is just a sketch of the shape I expect, not the actual answer.

Quick reminder: use the exact identifiers from the instance input — don’t rename them and don’t invent new labels. Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".","{'distance_matrix': [[0, 23, 39, 34, 25], [23, 0, 47, 38, 36], [39, 47, 0, 29, 48], [34, 38, 29, 0, 29], [25, 36, 48, 29, 0]], 'p': 1, 'objective': 38.0}","{'facilities': [3], 'assignments': [3, 3, 3, 3, 3]}",38.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 23, 'C': 39, 'D': 34, 'E': 25}}, {'id': 'B', 'distances': {'A': 23, 'B': 0, 'C': 47, 'D': 38, 'E': 36}}, {'id': 'C', 'distances': {'A': 39, 'B': 47, 'C': 0, 'D': 29, 'E': 48}}, {'id': 'D', 'distances': {'A': 34, 'B': 38, 'C': 29, 'D': 0, 'E': 29}}, {'id': 'E', 'distances': {'A': 25, 'B': 36, 'C': 48, 'D': 29, 'E': 0}}], 'objective': 38.0}","{'selected': ['D'], 'assignments': ['D', 'D', 'D', 'D', 'D']}",40,csv,names
PCENTER,PCENTER,"Many people in outlying hamlets depend on a few pickup boxes, so the planners must choose a set number of box locations from candidate sites and assign every delivery point to one chosen box only. The measure of a good setup is straightforward: find the house that ends up farthest from its box, note that distance, and aim to make that farthest distance as small as possible. The specific site options and the travel distances are listed below.

{
  ""total_locations"": 5,
  ""boxes_to_open"": 1,
  ""location_ids"": [
    1,
    2,
    3,
    4,
    5
  ],
  ""data"": [
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 2,
      ""travel_distance"": 25
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 3,
      ""travel_distance"": 36
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 4,
      ""travel_distance"": 37
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 5,
      ""travel_distance"": 22
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 1,
      ""travel_distance"": 25
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 3,
      ""travel_distance"": 36
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 4,
      ""travel_distance"": 30
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 5,
      ""travel_distance"": 28
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 1,
      ""travel_distance"": 36
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 2,
      ""travel_distance"": 36
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 4,
      ""travel_distance"": 37
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 5,
      ""travel_distance"": 22
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 1,
      ""travel_distance"": 37
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 2,
      ""travel_distance"": 30
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 3,
      ""travel_distance"": 37
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 5,
      ""travel_distance"": 22
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 1,
      ""travel_distance"": 22
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 2,
      ""travel_distance"": 28
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 3,
      ""travel_distance"": 22
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 4,
      ""travel_distance"": 22
    }
  ]
}

If you'd like to send a proposed setup, just drop it in a little JSON snippet like this so I can read it easily:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

This just means: ""selected"" is the list of box sites you plan to open, and ""assignments"" says, for each delivery location in the instance, which opened site it's assigned to. Super casual sketch — not the final answer, just the shape I expect.

Also, please use the exact identifiers from the instance input — don't rename them or invent new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 25, 36, 37, 22], [25, 0, 36, 30, 28], [36, 36, 0, 37, 22], [37, 30, 37, 0, 22], [22, 28, 22, 22, 0]], 'p': 1, 'objective': 28.0}","{'facilities': [4], 'assignments': [4, 4, 4, 4, 4]}",28.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 25, 3: 36, 4: 37, 5: 22}}, {'id': 2, 'distances': {1: 25, 2: 0, 3: 36, 4: 30, 5: 28}}, {'id': 3, 'distances': {1: 36, 2: 36, 3: 0, 4: 37, 5: 22}}, {'id': 4, 'distances': {1: 37, 2: 30, 3: 37, 4: 0, 5: 22}}, {'id': 5, 'distances': {1: 22, 2: 28, 3: 22, 4: 22, 5: 0}}], 'objective': 28.0}","{'selected': [5], 'assignments': [5, 5, 5, 5, 5]}",41,json,1
PCENTER,PCENTER,"I’ve been thinking about the little library van route: pick a fixed number of mobile book stops around town, then for each neighborhood cluster decide which one of those stops it will use. The idea is to arrange things so the person who has to walk the farthest still has as short a trip as possible — for any plan, look at every neighborhood’s travel distance to its assigned stop and take the largest one, and the goal is to make that largest distance as small as it can be. Every neighborhood has to be assigned to exactly one chosen stop — nothing left out and no neighborhood sent to two different stops. The exact candidate stops and the distances between places are listed below.

{
  ""total_candidate_locations"": 5,
  ""num_stops_to_open"": 1,
  ""location_identifiers"": [
    1,
    2,
    3,
    4,
    5
  ],
  ""data"": [
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 2,
      ""travel_distance"": 33
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 3,
      ""travel_distance"": 34
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 4,
      ""travel_distance"": 23
    },
    {
      ""origin_location_id"": 1,
      ""destination_location_id"": 5,
      ""travel_distance"": 22
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 1,
      ""travel_distance"": 33
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 3,
      ""travel_distance"": 37
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 4,
      ""travel_distance"": 19
    },
    {
      ""origin_location_id"": 2,
      ""destination_location_id"": 5,
      ""travel_distance"": 34
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 1,
      ""travel_distance"": 34
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 2,
      ""travel_distance"": 37
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 4,
      ""travel_distance"": 33
    },
    {
      ""origin_location_id"": 3,
      ""destination_location_id"": 5,
      ""travel_distance"": 28
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 1,
      ""travel_distance"": 23
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 2,
      ""travel_distance"": 19
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 3,
      ""travel_distance"": 33
    },
    {
      ""origin_location_id"": 4,
      ""destination_location_id"": 5,
      ""travel_distance"": 23
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 1,
      ""travel_distance"": 22
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 2,
      ""travel_distance"": 34
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 3,
      ""travel_distance"": 28
    },
    {
      ""origin_location_id"": 5,
      ""destination_location_id"": 4,
      ""travel_distance"": 23
    }
  ]
}

Oh, and to keep things machine-friendly, here's the little JSON layout I'd like the final plan to follow — nothing fancy, just a tidy shape:

{
  ""solution"": {
    ""selected"": [<stop_to_open>, <stop_to_open>, ...],
    ""assignments"": [<assigned_stop>, <assigned_stop>, ...]
  }
}

""selected"" is where you list which stops (by their location IDs) you're opening, and ""assignments"" says, for each neighborhood in the same order as the input, which opened stop it will use. Think of it like filling out a short form: pick the stops, then say who goes to which stop.

This is just a sketch of the expected shape, not the actual answer — I'll fill in the real IDs and assignments based on the instance details.

Please make sure to use the exact identifiers from the instance input — do not rename them or invent new labels. Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.","{'distance_matrix': [[0, 33, 34, 23, 22], [33, 0, 37, 19, 34], [34, 37, 0, 33, 28], [23, 19, 33, 0, 23], [22, 34, 28, 23, 0]], 'p': 1, 'objective': 33.0}","{'facilities': [3], 'assignments': [3, 3, 3, 3, 3]}",33.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 33, 3: 34, 4: 23, 5: 22}}, {'id': 2, 'distances': {1: 33, 2: 0, 3: 37, 4: 19, 5: 34}}, {'id': 3, 'distances': {1: 34, 2: 37, 3: 0, 4: 33, 5: 28}}, {'id': 4, 'distances': {1: 23, 2: 19, 3: 33, 4: 0, 5: 23}}, {'id': 5, 'distances': {1: 22, 2: 34, 3: 28, 4: 23, 5: 0}}], 'objective': 33.0}","{'selected': [4], 'assignments': [4, 4, 4, 4, 4]}",42,json,1
PCENTER,PCENTER,"Out at the coastal camping area the team has to pick a fixed number of restroom sites from the available locations and tie every campsite to one of the chosen restrooms. Every campsite gets assigned to one and only one opened restroom, and the planners have measured the distance between every pair of spots. For any choice, measure the walk for each campsite to its assigned restroom, then look at the worst — the longest walk anyone has to make — and aim for the plan where that worst walk is the shortest it can be. The concrete locations and distances are provided below.

There are 7 distinct spots; the team must open 1 restrooms, and the site identifiers are 1, 2, 3, 4, 5, 6, 7.
Measured walking distance from 1 to 2 is 52.
Measured walking distance from 1 to 3 is 64.
Measured walking distance from 1 to 4 is 79.
Measured walking distance from 1 to 5 is 54.
Measured walking distance from 1 to 6 is 78.
Measured walking distance from 1 to 7 is 58.
Measured walking distance from 2 to 1 is 52.
Measured walking distance from 2 to 3 is 62.
Measured walking distance from 2 to 4 is 77.
Measured walking distance from 2 to 5 is 23.
Measured walking distance from 2 to 6 is 63.
Measured walking distance from 2 to 7 is 70.
Measured walking distance from 3 to 1 is 64.
Measured walking distance from 3 to 2 is 62.
Measured walking distance from 3 to 4 is 15.
Measured walking distance from 3 to 5 is 75.
Measured walking distance from 3 to 6 is 53.
Measured walking distance from 3 to 7 is 57.
Measured walking distance from 4 to 1 is 79.
Measured walking distance from 4 to 2 is 77.
Measured walking distance from 4 to 3 is 15.
Measured walking distance from 4 to 5 is 78.
Measured walking distance from 4 to 6 is 38.
Measured walking distance from 4 to 7 is 72.
Measured walking distance from 5 to 1 is 54.
Measured walking distance from 5 to 2 is 23.
Measured walking distance from 5 to 3 is 75.
Measured walking distance from 5 to 4 is 78.
Measured walking distance from 5 to 6 is 40.
Measured walking distance from 5 to 7 is 54.
Measured walking distance from 6 to 1 is 78.
Measured walking distance from 6 to 2 is 63.
Measured walking distance from 6 to 3 is 53.
Measured walking distance from 6 to 4 is 38.
Measured walking distance from 6 to 5 is 40.
Measured walking distance from 6 to 7 is 92.
Measured walking distance from 7 to 1 is 58.
Measured walking distance from 7 to 2 is 70.
Measured walking distance from 7 to 3 is 57.
Measured walking distance from 7 to 4 is 72.
Measured walking distance from 7 to 5 is 54.
Measured walking distance from 7 to 6 is 92.
These distances are to be used to assign each campsite to an opened restroom and to select the plan that minimizes the maximum walk.

If you want to send back the plan, just use this simple JSON shape so it's easy to read and parse — something like:

{
  ""solution"": {
    ""selected"": [<restroom_to_open>, <restroom_to_open>, ...],
    ""assignments"": [<chosen_open_restroom>, <chosen_open_restroom>, ...]
  }
}

""selected"" is where you list the restroom sites you decide to open. ""assignments"" shows, in the same order as the campsite list from the instance, which opened restroom each campsite is tied to. This is just the shape of the reply I expect — a sketch, not the actual final plan.

Please be sure to use the exact identifiers from the instance input when you fill this in — do not rename or invent new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 52, 64, 79, 54, 78, 58], [52, 0, 62, 77, 23, 63, 70], [64, 62, 0, 15, 75, 53, 57], [79, 77, 15, 0, 78, 38, 72], [54, 23, 75, 78, 0, 40, 54], [78, 63, 53, 38, 40, 0, 92], [58, 70, 57, 72, 54, 92, 0]], 'p': 1, 'objective': 75.0}","{'facilities': [2], 'assignments': [2, 2, 2, 2, 2, 2, 2]}",75.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 52, 3: 64, 4: 79, 5: 54, 6: 78, 7: 58}}, {'id': 2, 'distances': {1: 52, 2: 0, 3: 62, 4: 77, 5: 23, 6: 63, 7: 70}}, {'id': 3, 'distances': {1: 64, 2: 62, 3: 0, 4: 15, 5: 75, 6: 53, 7: 57}}, {'id': 4, 'distances': {1: 79, 2: 77, 3: 15, 4: 0, 5: 78, 6: 38, 7: 72}}, {'id': 5, 'distances': {1: 54, 2: 23, 3: 75, 4: 78, 5: 0, 6: 40, 7: 54}}, {'id': 6, 'distances': {1: 78, 2: 63, 3: 53, 4: 38, 5: 40, 6: 0, 7: 92}}, {'id': 7, 'distances': {1: 58, 2: 70, 3: 57, 4: 72, 5: 54, 6: 92, 7: 0}}], 'objective': 75.0}","{'selected': [3], 'assignments': [3, 3, 3, 3, 3, 3, 3]}",43,nl,1
PCENTER,PCENTER,"We need to figure out where to install a set number of vending machines and which dorms and teaching areas should use each one. Every dorm or classroom must be assigned to one, and only one, chosen machine — nothing can be skipped or served twice. To see how good a setup is, measure each assigned walking distance and then take the largest distance anyone would have to walk; the goal is to make that biggest walk as small as it can be. The specific instance details follow below.

There are 8 candidate sites in total, we must install 1 vending machines, and the site identifiers are 0, 1, 2, 3, 4, 5, 6, 7.
From 0 to 1 the walking distance is 41.
From 0 to 2 the walking distance is 35.
From 0 to 3 the walking distance is 31.
From 0 to 4 the walking distance is 38.
From 0 to 5 the walking distance is 24.
From 0 to 6 the walking distance is 28.
From 0 to 7 the walking distance is 31.
From 1 to 0 the walking distance is 41.
From 1 to 2 the walking distance is 67.
From 1 to 3 the walking distance is 40.
From 1 to 4 the walking distance is 53.
From 1 to 5 the walking distance is 51.
From 1 to 6 the walking distance is 37.
From 1 to 7 the walking distance is 65.
From 2 to 0 the walking distance is 35.
From 2 to 1 the walking distance is 67.
From 2 to 3 the walking distance is 50.
From 2 to 4 the walking distance is 51.
From 2 to 5 the walking distance is 34.
From 2 to 6 the walking distance is 38.
From 2 to 7 the walking distance is 55.
From 3 to 0 the walking distance is 31.
From 3 to 1 the walking distance is 40.
From 3 to 2 the walking distance is 50.
From 3 to 4 the walking distance is 57.
From 3 to 5 the walking distance is 37.
From 3 to 6 the walking distance is 23.
From 3 to 7 the walking distance is 54.
From 4 to 0 the walking distance is 38.
From 4 to 1 the walking distance is 53.
From 4 to 2 the walking distance is 51.
From 4 to 3 the walking distance is 57.
From 4 to 5 the walking distance is 47.
From 4 to 6 the walking distance is 38.
From 4 to 7 the walking distance is 46.
From 5 to 0 the walking distance is 24.
From 5 to 1 the walking distance is 51.
From 5 to 2 the walking distance is 34.
From 5 to 3 the walking distance is 37.
From 5 to 4 the walking distance is 47.
From 5 to 6 the walking distance is 14.
From 5 to 7 the walking distance is 45.
From 6 to 0 the walking distance is 28.
From 6 to 1 the walking distance is 37.
From 6 to 2 the walking distance is 38.
From 6 to 3 the walking distance is 23.
From 6 to 4 the walking distance is 38.
From 6 to 5 the walking distance is 14.
From 6 to 7 the walking distance is 49.
From 7 to 0 the walking distance is 31.
From 7 to 1 the walking distance is 65.
From 7 to 2 the walking distance is 55.
From 7 to 3 the walking distance is 54.
From 7 to 4 the walking distance is 46.
From 7 to 5 the walking distance is 45.
From 7 to 6 the walking distance is 49.
We will use these pairwise walking distances to assign every location and evaluate the maximum walk under the 1-machine requirement.

I’ll show the shape I want the answer in as a little JSON sketch so it’s clear what to fill in — nothing fancy, just follow this layout.

{
  ""solution"": {
    ""selected"": [<machine_to_open>, <machine_to_open>, ...],
    ""assignments"": [<chosen_machine_location>, <chosen_machine_location>, ...]
  }
}

""selected"" is where you list the locations you open vending machines (one identifier per machine). ""assignments"" is a list, in the same order as the instance’s locations, saying which opened machine that location will use. Think of it like a simple form: which machines are open, and then for each dorm/area which machine they’re assigned to.

This JSON is just a sketch of the expected shape, not the actual answer. Also, don’t rename any identifiers from the instance input — use them exactly as given.
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 41, 35, 31, 38, 24, 28, 31], [41, 0, 67, 40, 53, 51, 37, 65], [35, 67, 0, 50, 51, 34, 38, 55], [31, 40, 50, 0, 57, 37, 23, 54], [38, 53, 51, 57, 0, 47, 38, 46], [24, 51, 34, 37, 47, 0, 14, 45], [28, 37, 38, 23, 38, 14, 0, 49], [31, 65, 55, 54, 46, 45, 49, 0]], 'p': 1, 'objective': 41.0}","{'facilities': [0], 'assignments': [0, 0, 0, 0, 0, 0, 0, 0]}",41.0,"{'problem_type': 'PCENTER', 'num_nodes': 8, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 41, 2: 35, 3: 31, 4: 38, 5: 24, 6: 28, 7: 31}}, {'id': 1, 'distances': {0: 41, 1: 0, 2: 67, 3: 40, 4: 53, 5: 51, 6: 37, 7: 65}}, {'id': 2, 'distances': {0: 35, 1: 67, 2: 0, 3: 50, 4: 51, 5: 34, 6: 38, 7: 55}}, {'id': 3, 'distances': {0: 31, 1: 40, 2: 50, 3: 0, 4: 57, 5: 37, 6: 23, 7: 54}}, {'id': 4, 'distances': {0: 38, 1: 53, 2: 51, 3: 57, 4: 0, 5: 47, 6: 38, 7: 46}}, {'id': 5, 'distances': {0: 24, 1: 51, 2: 34, 3: 37, 4: 47, 5: 0, 6: 14, 7: 45}}, {'id': 6, 'distances': {0: 28, 1: 37, 2: 38, 3: 23, 4: 38, 5: 14, 6: 0, 7: 49}}, {'id': 7, 'distances': {0: 31, 1: 65, 2: 55, 3: 54, 4: 46, 5: 45, 6: 49, 7: 0}}], 'objective': 41.0}","{'selected': [0], 'assignments': [0, 0, 0, 0, 0, 0, 0, 0]}",44,nl,0
PCENTER,PCENTER,"Someone’s been asked to place a limited number of potable-water kiosks around a rural district and then link every small settlement to one of those kiosks. The setup is strict: only the allotted number of kiosks can be opened, and each hamlet must be attached to a single kiosk—no hamlet can be skipped or served twice. The plan that’s best is the one that keeps the biggest distance any hamlet must travel as small as possible, meaning after assignments are made, the single largest hamlet-to-kiosk distance should be as low as it can go. The specific sites and distances are shown below.

There are 6 candidate locations named A, B, C, D, E, F, and exactly 1 kiosks must be opened among them.
From A to B the travel distance is 36.
From A to C the travel distance is 4.
From A to D the travel distance is 34.
From A to E the travel distance is 29.
From A to F the travel distance is 30.
From B to A the travel distance is 36.
From B to C the travel distance is 37.
From B to D the travel distance is 34.
From B to E the travel distance is 51.
From B to F the travel distance is 41.
From C to A the travel distance is 4.
From C to B the travel distance is 37.
From C to D the travel distance is 33.
From C to E the travel distance is 33.
From C to F the travel distance is 32.
From D to A the travel distance is 34.
From D to B the travel distance is 34.
From D to C the travel distance is 33.
From D to E the travel distance is 43.
From D to F the travel distance is 38.
From E to A the travel distance is 29.
From E to B the travel distance is 51.
From E to C the travel distance is 33.
From E to D the travel distance is 43.
From E to F the travel distance is 46.
From F to A the travel distance is 30.
From F to B the travel distance is 41.
From F to C the travel distance is 32.
From F to D the travel distance is 38.
From F to E the travel distance is 46.
Use these distance entries to assign every hamlet to a single opened kiosk so the largest hamlet-to-kiosk distance is as small as possible.

When you report the choices and who goes to which kiosk, it's easiest if you stick to a simple JSON layout like this:

{
  ""solution"": {
    ""selected"": [""<site_to_open>"", ""<site_to_open>"", ...],
    ""assignments"": [""<chosen_open_site>"", ""<chosen_open_site>"", ...]
  }
}

This just sketches the shape I expect: ""selected"" is the list of kiosk sites you decide to open, and ""assignments"" lists, for each settlement in the same order as the input, which opened site it gets linked to. Think of it like a quick form — which kiosks are turned on, and who is assigned to each.

This JSON is only a template of the expected format, not the final answer. Please make sure all identifiers you use match exactly the ones in the instance input — no renaming, no invented labels. 

for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 36, 4, 34, 29, 30], [36, 0, 37, 34, 51, 41], [4, 37, 0, 33, 33, 32], [34, 34, 33, 0, 43, 38], [29, 51, 33, 43, 0, 46], [30, 41, 32, 38, 46, 0]], 'p': 1, 'objective': 36.0}","{'facilities': [0], 'assignments': [0, 0, 0, 0, 0, 0]}",36.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 36, 'C': 4, 'D': 34, 'E': 29, 'F': 30}}, {'id': 'B', 'distances': {'A': 36, 'B': 0, 'C': 37, 'D': 34, 'E': 51, 'F': 41}}, {'id': 'C', 'distances': {'A': 4, 'B': 37, 'C': 0, 'D': 33, 'E': 33, 'F': 32}}, {'id': 'D', 'distances': {'A': 34, 'B': 34, 'C': 33, 'D': 0, 'E': 43, 'F': 38}}, {'id': 'E', 'distances': {'A': 29, 'B': 51, 'C': 33, 'D': 43, 'E': 0, 'F': 46}}, {'id': 'F', 'distances': {'A': 30, 'B': 41, 'C': 32, 'D': 38, 'E': 46, 'F': 0}}], 'objective': 36.0}","{'selected': ['A'], 'assignments': ['A', 'A', 'A', 'A', 'A', 'A']}",45,nl,names
PCENTER,PCENTER,"Someone needs to pick a handful of docks to keep open around the neighborhood and then assign each popular origin to one of the open docks — no origin skipped and no origin linked to multiple docks. The trick is to choose the open docks so that, after assignments, the longest walk anyone ends up with (the single biggest distance among all assignments) is as small as it can be. The actual candidate sites and the travel distances between them are shown below.

{
  ""total_candidate_locations"": 6,
  ""docks_to_keep_open"": 1,
  ""location_ids"": [
    0,
    1,
    2,
    3,
    4,
    5
  ],
  ""data"": [
    {
      ""source_location_id"": 0,
      ""target_location_id"": 1,
      ""walking_distance"": 93
    },
    {
      ""source_location_id"": 0,
      ""target_location_id"": 2,
      ""walking_distance"": 111
    },
    {
      ""source_location_id"": 0,
      ""target_location_id"": 3,
      ""walking_distance"": 53
    },
    {
      ""source_location_id"": 0,
      ""target_location_id"": 4,
      ""walking_distance"": 63
    },
    {
      ""source_location_id"": 0,
      ""target_location_id"": 5,
      ""walking_distance"": 47
    },
    {
      ""source_location_id"": 1,
      ""target_location_id"": 0,
      ""walking_distance"": 93
    },
    {
      ""source_location_id"": 1,
      ""target_location_id"": 2,
      ""walking_distance"": 138
    },
    {
      ""source_location_id"": 1,
      ""target_location_id"": 3,
      ""walking_distance"": 56
    },
    {
      ""source_location_id"": 1,
      ""target_location_id"": 4,
      ""walking_distance"": 30
    },
    {
      ""source_location_id"": 1,
      ""target_location_id"": 5,
      ""walking_distance"": 50
    },
    {
      ""source_location_id"": 2,
      ""target_location_id"": 0,
      ""walking_distance"": 111
    },
    {
      ""source_location_id"": 2,
      ""target_location_id"": 1,
      ""walking_distance"": 138
    },
    {
      ""source_location_id"": 2,
      ""target_location_id"": 3,
      ""walking_distance"": 158
    },
    {
      ""source_location_id"": 2,
      ""target_location_id"": 4,
      ""walking_distance"": 165
    },
    {
      ""source_location_id"": 2,
      ""target_location_id"": 5,
      ""walking_distance"": 152
    },
    {
      ""source_location_id"": 3,
      ""target_location_id"": 0,
      ""walking_distance"": 53
    },
    {
      ""source_location_id"": 3,
      ""target_location_id"": 1,
      ""walking_distance"": 56
    },
    {
      ""source_location_id"": 3,
      ""target_location_id"": 2,
      ""walking_distance"": 158
    },
    {
      ""source_location_id"": 3,
      ""target_location_id"": 4,
      ""walking_distance"": 26
    },
    {
      ""source_location_id"": 3,
      ""target_location_id"": 5,
      ""walking_distance"": 10
    },
    {
      ""source_location_id"": 4,
      ""target_location_id"": 0,
      ""walking_distance"": 63
    },
    {
      ""source_location_id"": 4,
      ""target_location_id"": 1,
      ""walking_distance"": 30
    },
    {
      ""source_location_id"": 4,
      ""target_location_id"": 2,
      ""walking_distance"": 165
    },
    {
      ""source_location_id"": 4,
      ""target_location_id"": 3,
      ""walking_distance"": 26
    },
    {
      ""source_location_id"": 4,
      ""target_location_id"": 5,
      ""walking_distance"": 20
    },
    {
      ""source_location_id"": 5,
      ""target_location_id"": 0,
      ""walking_distance"": 47
    },
    {
      ""source_location_id"": 5,
      ""target_location_id"": 1,
      ""walking_distance"": 50
    },
    {
      ""source_location_id"": 5,
      ""target_location_id"": 2,
      ""walking_distance"": 152
    },
    {
      ""source_location_id"": 5,
      ""target_location_id"": 3,
      ""walking_distance"": 10
    },
    {
      ""source_location_id"": 5,
      ""target_location_id"": 4,
      ""walking_distance"": 20
    }
  ]
}

Oh, and when you reply with the chosen docks and who goes to which dock, please follow this simple JSON sketch for the shape of the answer — nothing fancy, just that layout.

{
  ""solution"": {
    ""selected"": [<dock_to_open>, <dock_to_open>, ...],
    ""assignments"": [<chosen_open_dock>, <chosen_open_dock>, ...]
  }
}

This is just a friendly template: ""selected"" is the list of docks we keep open, and ""assignments"" lists, for each origin in the same order as the instance, which open dock it's assigned to. Think of it like filling out a short form — first pick the open docks, then for every origin write down which open dock it uses.

Note that the JSON above is only the expected shape — not the final answer itself. Also, please use the exact identifiers from the instance input; do not rename them or invent new labels.

For example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 93, 111, 53, 63, 47], [93, 0, 138, 56, 30, 50], [111, 138, 0, 158, 165, 152], [53, 56, 158, 0, 26, 10], [63, 30, 165, 26, 0, 20], [47, 50, 152, 10, 20, 0]], 'p': 1, 'objective': 111.0}","{'facilities': [0], 'assignments': [0, 0, 0, 0, 0, 0]}",111.0,"{'problem_type': 'PCENTER', 'num_nodes': 6, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 93, 2: 111, 3: 53, 4: 63, 5: 47}}, {'id': 1, 'distances': {0: 93, 1: 0, 2: 138, 3: 56, 4: 30, 5: 50}}, {'id': 2, 'distances': {0: 111, 1: 138, 2: 0, 3: 158, 4: 165, 5: 152}}, {'id': 3, 'distances': {0: 53, 1: 56, 2: 158, 3: 0, 4: 26, 5: 10}}, {'id': 4, 'distances': {0: 63, 1: 30, 2: 165, 3: 26, 4: 0, 5: 20}}, {'id': 5, 'distances': {0: 47, 1: 50, 2: 152, 3: 10, 4: 20, 5: 0}}], 'objective': 111.0}","{'selected': [0], 'assignments': [0, 0, 0, 0, 0, 0]}",46,json,0
PCENTER,PCENTER,"Around here the challenge is to pick only a certain number of approved donation-bin locations and then connect each apartment cluster to one of them. The whole point is to keep the worst walk short: after picking the bins and assigning clusters, look at every cluster’s distance to its bin, pick the farthest distance, and try to minimize that farthest trip. Each cluster must be assigned to exactly one selected bin, with no overlaps or omissions. The concrete locations and distances follow below.

There are 5 distinct locations, you must open exactly 1 of the approved spots, and the location identifiers are A, B, C, D, E.
From A to B the walking distance is 32.
From A to C the walking distance is 26.
From A to D the walking distance is 9.
From A to E the walking distance is 29.
From B to A the walking distance is 32.
From B to C the walking distance is 8.
From B to D the walking distance is 25.
From B to E the walking distance is 32.
From C to A the walking distance is 26.
From C to B the walking distance is 8.
From C to D the walking distance is 18.
From C to E the walking distance is 27.
From D to A the walking distance is 9.
From D to B the walking distance is 25.
From D to C the walking distance is 18.
From D to E the walking distance is 27.
From E to A the walking distance is 29.
From E to B the walking distance is 32.
From E to C the walking distance is 27.
From E to D the walking distance is 27.
Use these distances to assign each cluster to one chosen bin so the maximum assigned walk is as small as possible.

Also, when you send back the chosen plan, please use this simple JSON shape so it's easy to check automatically — nothing fancy, just the two pieces we need.

{
  ""solution"": {
    ""selected"": [<bin_to_open>, <bin_to_open>, ...],
    ""assignments"": [<assigned_bin>, <assigned_bin>, ...]
  }
}

Here ""selected"" is the short list of donation-bin sites you decide to open, and ""assignments"" lists, for every apartment cluster in the same order as the instance, which opened bin that cluster is assigned to. Think of it like filling out a form: pick the bins, then for each cluster write which bin it uses.

This JSON is just a sketch of the shape I expect — not the final answers themselves.

Please be sure to use the exact identifiers from the instance input, with no renaming and no new labels. For example: ""Valid identifiers look like plain numbers such as ""1"" or ""23"", single capital letters like ""A"" or ""B"", or a capital letter followed by digits like ""A1"" or ""X7"".""","{'distance_matrix': [[0, 32, 26, 9, 29], [32, 0, 8, 25, 32], [26, 8, 0, 18, 27], [9, 25, 18, 0, 27], [29, 32, 27, 27, 0]], 'p': 1, 'objective': 27.0}","{'facilities': [2], 'assignments': [2, 2, 2, 2, 2]}",27.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 32, 'C': 26, 'D': 9, 'E': 29}}, {'id': 'B', 'distances': {'A': 32, 'B': 0, 'C': 8, 'D': 25, 'E': 32}}, {'id': 'C', 'distances': {'A': 26, 'B': 8, 'C': 0, 'D': 18, 'E': 27}}, {'id': 'D', 'distances': {'A': 9, 'B': 25, 'C': 18, 'D': 0, 'E': 27}}, {'id': 'E', 'distances': {'A': 29, 'B': 32, 'C': 27, 'D': 27, 'E': 0}}], 'objective': 27.0}","{'selected': ['C'], 'assignments': ['C', 'C', 'C', 'C', 'C']}",47,markdown_table,names
PCENTER,PCENTER,"I was helping a team plan a series of weekend pop-ups around town: there’s a fixed number of booths that can be opened, and every neighborhood cluster needs to be handed off to exactly one of those booths — no cluster can be left out or split between two booths. The idea is to pick which venues to open so that the single longest walk any neighborhood has to make to its assigned booth is as short as possible — in other words, check the cluster that ends up farthest from its booth and make that distance as small as it can be. The concrete list of possible spots and the travel distances between places are shown below.

# total_locations=5
# num_booths_to_open=1
# location_ids=0, 1, 2, 3, 4
from_location_id,to_location_id,walking_distance
0,1,120
0,2,92
0,3,75
0,4,91
1,0,120
1,2,85
1,3,106
1,4,84
2,0,92
2,1,85
2,3,89
2,4,93
3,0,75
3,1,106
3,2,89
3,4,97
4,0,91
4,1,84
4,2,93
4,3,97

If you want to hand me the final choices, just use this little JSON shape so it's easy to parse — nothing fancy, just the two parts we need:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

Think of it like a short form: ""selected"" is the list of spots you decide to open for the pop-ups, and ""assignments"" is, in the same order as the input locations, which opened spot each one hooks up to. Super casual — just fill those placeholders with the exact IDs from the instance when you give the real answer.

This JSON is only a sketch of the shape I expect, not the actual answer.

Please be sure all identifiers match the instance input exactly — don't rename them or invent new labels. For example:
- Valid identifiers look like plain numbers such as “1” or “23”
- single capital letters like “A” or “B”
- a capital letter followed by digits like “A1” or “X7”","{'distance_matrix': [[0, 120, 92, 75, 91], [120, 0, 85, 106, 84], [92, 85, 0, 89, 93], [75, 106, 89, 0, 97], [91, 84, 93, 97, 0]], 'p': 1, 'objective': 93.0}","{'facilities': [2], 'assignments': [2, 2, 2, 2, 2]}",93.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 0, 'distances': {0: 0, 1: 120, 2: 92, 3: 75, 4: 91}}, {'id': 1, 'distances': {0: 120, 1: 0, 2: 85, 3: 106, 4: 84}}, {'id': 2, 'distances': {0: 92, 1: 85, 2: 0, 3: 89, 4: 93}}, {'id': 3, 'distances': {0: 75, 1: 106, 2: 89, 3: 0, 4: 97}}, {'id': 4, 'distances': {0: 91, 1: 84, 2: 93, 3: 97, 4: 0}}], 'objective': 93.0}","{'selected': [2], 'assignments': [2, 2, 2, 2, 2]}",48,csv,0
PCENTER,PCENTER,"On a busy weekend the team needs to decide which aisles will host a preset number of pickup counters; distances between all candidate spots are known. Each product section must be linked to one open counter — nothing can be left unassigned or attached to more than one counter. To pick the best arrangement, look at the distance every section’s customers would walk to their counter, pick out the largest of those distances, and choose the arrangement that makes that largest walk as small as possible. The detailed map and distances follow below.

{
  ""num_candidate_aisles"": 7,
  ""num_counters_to_open"": 1,
  ""candidate_spot_ids"": [
    1,
    2,
    3,
    4,
    5,
    6,
    7
  ],
  ""data"": [
    {
      ""from_spot_id"": 1,
      ""to_spot_id"": 2,
      ""walking_distance"": 36
    },
    {
      ""from_spot_id"": 1,
      ""to_spot_id"": 3,
      ""walking_distance"": 46
    },
    {
      ""from_spot_id"": 1,
      ""to_spot_id"": 4,
      ""walking_distance"": 47
    },
    {
      ""from_spot_id"": 1,
      ""to_spot_id"": 5,
      ""walking_distance"": 41
    },
    {
      ""from_spot_id"": 1,
      ""to_spot_id"": 6,
      ""walking_distance"": 38
    },
    {
      ""from_spot_id"": 1,
      ""to_spot_id"": 7,
      ""walking_distance"": 38
    },
    {
      ""from_spot_id"": 2,
      ""to_spot_id"": 1,
      ""walking_distance"": 36
    },
    {
      ""from_spot_id"": 2,
      ""to_spot_id"": 3,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 2,
      ""to_spot_id"": 4,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 2,
      ""to_spot_id"": 5,
      ""walking_distance"": 30
    },
    {
      ""from_spot_id"": 2,
      ""to_spot_id"": 6,
      ""walking_distance"": 25
    },
    {
      ""from_spot_id"": 2,
      ""to_spot_id"": 7,
      ""walking_distance"": 9
    },
    {
      ""from_spot_id"": 3,
      ""to_spot_id"": 1,
      ""walking_distance"": 46
    },
    {
      ""from_spot_id"": 3,
      ""to_spot_id"": 2,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 3,
      ""to_spot_id"": 4,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 3,
      ""to_spot_id"": 5,
      ""walking_distance"": 24
    },
    {
      ""from_spot_id"": 3,
      ""to_spot_id"": 6,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 3,
      ""to_spot_id"": 7,
      ""walking_distance"": 27
    },
    {
      ""from_spot_id"": 4,
      ""to_spot_id"": 1,
      ""walking_distance"": 47
    },
    {
      ""from_spot_id"": 4,
      ""to_spot_id"": 2,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 4,
      ""to_spot_id"": 3,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 4,
      ""to_spot_id"": 5,
      ""walking_distance"": 29
    },
    {
      ""from_spot_id"": 4,
      ""to_spot_id"": 6,
      ""walking_distance"": 32
    },
    {
      ""from_spot_id"": 4,
      ""to_spot_id"": 7,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 5,
      ""to_spot_id"": 1,
      ""walking_distance"": 41
    },
    {
      ""from_spot_id"": 5,
      ""to_spot_id"": 2,
      ""walking_distance"": 30
    },
    {
      ""from_spot_id"": 5,
      ""to_spot_id"": 3,
      ""walking_distance"": 24
    },
    {
      ""from_spot_id"": 5,
      ""to_spot_id"": 4,
      ""walking_distance"": 29
    },
    {
      ""from_spot_id"": 5,
      ""to_spot_id"": 6,
      ""walking_distance"": 30
    },
    {
      ""from_spot_id"": 5,
      ""to_spot_id"": 7,
      ""walking_distance"": 33
    },
    {
      ""from_spot_id"": 6,
      ""to_spot_id"": 1,
      ""walking_distance"": 38
    },
    {
      ""from_spot_id"": 6,
      ""to_spot_id"": 2,
      ""walking_distance"": 25
    },
    {
      ""from_spot_id"": 6,
      ""to_spot_id"": 3,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 6,
      ""to_spot_id"": 4,
      ""walking_distance"": 32
    },
    {
      ""from_spot_id"": 6,
      ""to_spot_id"": 5,
      ""walking_distance"": 30
    },
    {
      ""from_spot_id"": 6,
      ""to_spot_id"": 7,
      ""walking_distance"": 22
    },
    {
      ""from_spot_id"": 7,
      ""to_spot_id"": 1,
      ""walking_distance"": 38
    },
    {
      ""from_spot_id"": 7,
      ""to_spot_id"": 2,
      ""walking_distance"": 9
    },
    {
      ""from_spot_id"": 7,
      ""to_spot_id"": 3,
      ""walking_distance"": 27
    },
    {
      ""from_spot_id"": 7,
      ""to_spot_id"": 4,
      ""walking_distance"": 28
    },
    {
      ""from_spot_id"": 7,
      ""to_spot_id"": 5,
      ""walking_distance"": 33
    },
    {
      ""from_spot_id"": 7,
      ""to_spot_id"": 6,
      ""walking_distance"": 22
    }
  ]
}

Also, to keep things easy to parse, please lay out the solution in this simple JSON shape when you reply:

{
  ""solution"": {
    ""selected"": [<aisle_to_open>, <aisle_to_open>, ...],
    ""assignments"": [<chosen_open_aisle>, <chosen_open_aisle>, ...]
  }
}

Think of ""selected"" as the list of aisles where we actually put pickup counters, and ""assignments"" as which open aisle each product section is tied to. Super casual — just a quick form showing which aisles are open and who goes to which one. This is just the expected shape, not the final plan itself.

Remember to use the exact identifiers from the instance input — don't rename them or invent new labels. 
- for example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 36, 46, 47, 41, 38, 38], [36, 0, 28, 28, 30, 25, 9], [46, 28, 0, 28, 24, 28, 27], [47, 28, 28, 0, 29, 32, 28], [41, 30, 24, 29, 0, 30, 33], [38, 25, 28, 32, 30, 0, 22], [38, 9, 27, 28, 33, 22, 0]], 'p': 1, 'objective': 36.0}","{'facilities': [1], 'assignments': [1, 1, 1, 1, 1, 1, 1]}",36.0,"{'problem_type': 'PCENTER', 'num_nodes': 7, 'num_open': 1, 'sites': [{'id': 1, 'distances': {1: 0, 2: 36, 3: 46, 4: 47, 5: 41, 6: 38, 7: 38}}, {'id': 2, 'distances': {1: 36, 2: 0, 3: 28, 4: 28, 5: 30, 6: 25, 7: 9}}, {'id': 3, 'distances': {1: 46, 2: 28, 3: 0, 4: 28, 5: 24, 6: 28, 7: 27}}, {'id': 4, 'distances': {1: 47, 2: 28, 3: 28, 4: 0, 5: 29, 6: 32, 7: 28}}, {'id': 5, 'distances': {1: 41, 2: 30, 3: 24, 4: 29, 5: 0, 6: 30, 7: 33}}, {'id': 6, 'distances': {1: 38, 2: 25, 3: 28, 4: 32, 5: 30, 6: 0, 7: 22}}, {'id': 7, 'distances': {1: 38, 2: 9, 3: 27, 4: 28, 5: 33, 6: 22, 7: 0}}], 'objective': 36.0}","{'selected': [2], 'assignments': [2, 2, 2, 2, 2, 2, 2]}",49,json,1
PCENTER,PCENTER,"There’s a constant puzzle in the terminal about which baggage belts to activate. Given a fixed number to turn on, every incoming gate must be tied to one of the active belts (no gate left out, no gate sent to two belts), and the way to tell if a choice is good is to find the passenger who walks the farthest under that plan. The goal is to keep that single worst walk as short as possible — measure it by the largest gate-to-carousel distance in the assignment. The specific layout and distances follow below.

This instance lists 5 candidate locations, requires activating 1 carousels, and provides the ordered location IDs A, B, C, D, E.
From A to B the walk distance is 52.
From A to C the walk distance is 66.
From A to D the walk distance is 62.
From A to E the walk distance is 71.
From B to A the walk distance is 52.
From B to C the walk distance is 37.
From B to D the walk distance is 43.
From B to E the walk distance is 49.
From C to A the walk distance is 66.
From C to B the walk distance is 37.
From C to D the walk distance is 60.
From C to E the walk distance is 65.
From D to A the walk distance is 62.
From D to B the walk distance is 43.
From D to C the walk distance is 60.
From D to E the walk distance is 53.
From E to A the walk distance is 71.
From E to B the walk distance is 49.
From E to C the walk distance is 65.
From E to D the walk distance is 53.
These distances are to be used to evaluate assignments and determine the maximum single walk given the 1 active carousels.

If you want to hand me a candidate plan, just drop it into this little JSON sketch so I can read it easily:

{
  ""solution"": {
    ""selected"": [<site_to_open>, <site_to_open>, ...],
    ""assignments"": [<chosen_open_site>, <chosen_open_site>, ...]
  }
}

This is just a simple form: ""selected"" is the list of carousels/belts you choose to switch on, and ""assignments"" is a list that, for each gate in the instance (in the instance order), names which of the chosen carousels that gate uses. Think of it like filling out a short checklist — which belts are active, and which belt each gate goes to. The JSON above is only a sketch of the shape I expect, not the actual solution.

Please be careful to use the exact identifiers from the instance input — don’t rename them and don’t invent new labels. For example: ""Valid identifiers look like plain numbers such as “1” or “23”, single capital letters like “A” or “B”, or a capital letter followed by digits like “A1” or “X7”.""","{'distance_matrix': [[0, 52, 66, 62, 71], [52, 0, 37, 43, 49], [66, 37, 0, 60, 65], [62, 43, 60, 0, 53], [71, 49, 65, 53, 0]], 'p': 1, 'objective': 52.0}","{'facilities': [1], 'assignments': [1, 1, 1, 1, 1]}",52.0,"{'problem_type': 'PCENTER', 'num_nodes': 5, 'num_open': 1, 'sites': [{'id': 'A', 'distances': {'A': 0, 'B': 52, 'C': 66, 'D': 62, 'E': 71}}, {'id': 'B', 'distances': {'A': 52, 'B': 0, 'C': 37, 'D': 43, 'E': 49}}, {'id': 'C', 'distances': {'A': 66, 'B': 37, 'C': 0, 'D': 60, 'E': 65}}, {'id': 'D', 'distances': {'A': 62, 'B': 43, 'C': 60, 'D': 0, 'E': 53}}, {'id': 'E', 'distances': {'A': 71, 'B': 49, 'C': 65, 'D': 53, 'E': 0}}], 'objective': 52.0}","{'selected': ['B'], 'assignments': ['B', 'B', 'B', 'B', 'B']}",50,markdown_table,names