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Given an undirected graph $(V,E)$ with weights on edges $\in Z$ is it possible to find a negative cycle with **even** length in polynomial time? |
Negative cycle of even length |
|graphs|graph-traversal|weighted-graphs| |
Given an undirected graph $(V,E)$ with weights on edges $\in Z$ is it possible to find a negative cycle of **even** length (not the weight of the cycle, but the number of edges contained in the cycle) in polynomial time? |
While this seems like homework, since it was asked 8 years ago it is probably past the point that the question asker can benefit from this for that course.
Build a graph $G_2 = (U, E_2)$ of where the vertices of $G_2$ are only the $u$-vertices from your original graph. And join an edge between any two tournaments $u... |
I have a very strange (arbitrary) problem that I have been trying to wrap my head around, but have been struggling to figure out how to accomplish this. I'll use Rust but in a pseudo-code fashion, these probably won't compile.
### Background
The library function takes a `&[u64]` (a slice/array/list of unsigned, 6... |
Splitting a Larger Unsigned Integer into Summable Parts |
|binary|bit-manipulation| |
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I was studying the dining philosopher problem and its solution using semaphores, and I came across this :
**Code for the ith philosopher:**
while(1)
{
take_chopstick[i];
take_chopstick[ (i+1) % 5] ;
EATING THE NOODLE
put_chopstick[i]... |
Order of releasing the semaphores |
|operating-systems|mutual-exclusion|critical-section|semaphore| |
I was studying the dining philosopher problem ([See problem description][1]) and its solution using semaphores, and I came across this :
**Code for the ith philosopher:**
while(1)
{
take_chopstick[i];
take_chopstick[ (i+1) % 5] ;
EATING THE NOODLE
... |
Given an undirected graph $(V,E)$ with weights on edges $\in Z$ is it possible to find a negative cycle of **even** length (not the weight of the cycle, but the number of edges contained in the cycle) in polynomial time?
Motivation:
-----------------
A simpler implementation of minimum weighted maximal matching ... |
What is the smallest 3-CNF system of equations possible that enforces the boolean expression: a = b + c for boolean variables a, b, c? |
what is the smallest 3-CNF possible that enforces the boolean expression: a = b + c? |
|boolean-algebra| |
What is the smallest 3-CNF system of equations possible that enforces the boolean expression: a = b + c for boolean variables a, b, c?
'Smallest' can be defined as: (1) Number of 3 CNF clauses. (2) Number of new variables introduced in the clauses.
We are interested in both cases. |
To begin with, note that $b+c$ can be expressed as the disjunction $b\lor c$. To enforce equation, you need to express "$a$ iff $(b\lor c)$", and that can encoded as the following CNF formula: $$ \phi = (\neg a \lor b \lor c) \wedge ((\neg b \wedge \neg c) \lor a)$$
Indeed, I used the fact that $A\to B \equiv \neg A... |
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To begin with, note that $b+c$ can be expressed as the disjunction $b\lor c$. To enforce equation, you need to express "$a$ iff $(b\lor c)$", and that can be encoded as the following CNF formula: $$ \phi = (\neg a \lor b \lor c) \wedge ((\neg b \wedge \neg c) \lor a)$$
Indeed, I used the fact that $A\to B \equiv \ne... |
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Given an undirected graph $(V,E)$ with weights on edges $\in Z$ is it possible to find a negative cycle of **even** length (not the weight of the cycle, but the number of edges contained in the cycle) in polynomial time?
Formally:
Find a sequence of vertices $\{v_1,v_2,...,v_k,v_1\}$ such that
1) For every $i$, ... |
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We have shown that any comparative sorting algorithm has the worst-case complexity of $\Omega(\log (n!)) = \Omega(n \log n)$, as it has to cover all the ways a permutation can be, and according to the pigeonhole principle, there is a case where more than or equal to half of the currently unknown permutations would be u... |
Is there a [non-comparative] sorting algorithm, sorting an array in $o(n \log n)$? |
|asymptotics|sorting| |
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Say I have the Halting Problem H = {<M,w> | M halts on input w},
and some undecidable Language L of the sort
L = {M | M does a thing}.
If I want to reduce H to L, from what I understand, I have to construct a Turing Machine M' which takes the input <M,w>, and if <M,w> halts, M' does the thing, if <M,w> doesn’t... |
Reducing from the complement of the Halting Problem |
|computability|reductions|halting-problem| |
Radix sort can sort an array in $O(n.w)$ where $w$ is the key length. As $w$ is $O(\log n)$, this means it can sort an array with the performance of $O(n \log n)$. In fact, every non-comparison sort in the form of $O(n.w)$ or $O(n.k/d)$ is $O(n \log n)$.
https://www.drdobbs.com/architecture-and-design/the-fastest-so... |
The term 'superexponential' is _not_ a precise description of algorithmic complexity, but it always suggests a growth greater than a simple exponential function of form $b^n, b > 1$. There are three main levels of superexponential growth:
1. Beyond the class __E__, the class of all problems with _simple exponential_... |
The term 'superexponential' is _not_ a precise description of algorithmic complexity, but it always suggests a growth greater in order than a simple exponential function of form $b^n$ for some $b > 1$. There are three main levels of superexponential growth:
1. Beyond the class __E__, the class of all problems with _... |
You have `n` types work items. Each work item must be distributed to at least ```m``` processors. There are k processors (n > k > m) available. You want to distribute work as fairly as possible.
Finally, you want to minimize the number of repeating pairs of work items across processors. I don't know how to express i... |
Distribute work minimizing |
|assignment-problem| |
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null |
Note that by reducing from the complement of the halting problem, that is from $\overline{HALT_{TM}} = \{ \langle M, w\rangle: \text{$M$ is a TM that does not halt on $w$} \}$, then you (by you I mean the reduction) need, upon getting in put of the form $\langle M, w\rangle$, to construct, halt and output a description... |
Note that by reducing from the complement of the halting problem, that is from $\overline{HALT_{TM}} = \{ \langle M, w\rangle: \text{$M$ is a TM that does not halt on $w$} \}$, then you (by you I mean the reduction) need, upon getting in put of the form $\langle M, w\rangle$, to construct, halt and output a description... |
For the question to be understandable, you should indicate that "a machine does a thing" actually means that "a machine performs some finite computation and then halts". Anyway, this should clarify the reduction concept to you:
Note that by reducing from the complement of the halting problem, that is from $\overline... |
**Context:**
While exploring the fundamentals of polyhedral optimization and **attempting to explore a connection map from the input Static Control Part (SCoP) to the iteration domain**, I am confused about whether I am making the correct connections or not.
**Explored connection map:**
[![enter image descript... |
Nested Loop (SCoP) to Iteration Domain in Polyhedral Optimization: Explored connection are correct? |
|optimization|sets|notation|loops|iteration| |
**Context:**
While exploring the fundamentals of polyhedral optimization and **attempting to explore a connection map from the input Static Control Part (SCoP) to the iteration domain**, I am confused about whether I am making the correct connections or not.
**Explored connection map:**
[![enter image descript... |
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**Context:**
While exploring the fundamentals of polyhedral optimization and **attempting to explore a connection from the input Static Control Part (SCoP) to the iteration domain from birds eye view**, I am confused about whether I am making the correct connections or not.
**Explored connection map:**
[![ente... |
Let $C_1$ and $C_2$ be languages representing the conditions $1$ and $2$, respectively. Then, $C_1 \in \text{NP-complete}$ and $C_2 \in \text{PTIME}$, and by writing down formally what you mentioned, we get $A = \overline{C_1} \cap \overline{C_2}$.
Now you can use the following claim:
**Claim:** A language $L$ i... |
Let $C_1$ and $C_2$ be languages representing the conditions $1$ and $2$, respectively. Then, $C_1 \in \text{NP-complete}$ and $C_2 \in \text{PTIME}$, and by writing down formally what you mentioned, we get $A = \overline{C_1} \cap \overline{C_2}$.
Now you can use the following claim:
**Claim:** A language $L$ i... |
The source hinted that it is by the definition of median and outlined how to verify an element in $O(1)$.
If $l > m$, $left = \lceil (l - m) / 2 \rceil$, $right = \lceil (l + m) / 2 \rceil$. Else, $left = 1$, $right = l$.
The median is guaranteed to be found in $A[left..right]$ using that range, or if not then tr... |
Consider the halting problem $HALT_{TM} = \{\langle M, w\rangle: M \text{ is a TM that halts on input } w\}$,
and some undecidable Language $L$ of the form $L = \{\langle M\rangle: M \text{ does a thing} \}$.
If I want to reduce the halting problem to $L$, from what I understand, I have to construct a Turing Mach... |
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It doesn't matter how you pop, either min first, max first, or alternating. It is still $O(\log n)$ for each pop. |
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Here's a counter example on [codeforces](https://codeforces.com/blog/entry/123083). However if it's *complete* bipartite matching, we can use the greedy Nobel winning [algorithm](https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley_algorithm) in Economics. |
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SCoP to Iteration Domain in Polyhedral Optimization: Explored connection are correct? |
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**Context:**
While exploring the fundamentals of polyhedral optimization and **attempting to explore a connection from the input Static Control Part (SCoP) to the iteration domain from birds eye view**, I am confused about whether I am making the correct connections or not.
**Explored connection map:**
[![ente... |
I have a very strange (arbitrary) problem that I have been trying to wrap my head around, but have been struggling to figure out how to accomplish this. I'll use Rust but in a pseudo-code fashion, these probably won't compile.
### Background
The library function takes a `&[u64]` (a slice/array/list of unsigned, 6... |
Having a complete bipartite graph with parts $A$ and $B$, which is edge-weighted, is there a way to compute a maximum weight matching such that:
1. Only a constant number $n$ of vertices from $A$ are used.
2. An edge points to each of the vertices in $B$.
In simple terms, *pruning* the original graph so that I g... |
Weighted bipartite perfect matching with a fixed number of vertices |
|graphs|combinatorics|matching|bipartite-matching|clustering| |
Given a Problem $A$ that has an answer $true$ if and only if both conditions $1$ and $2$ are $false$, for some conditions 1 and 2.
Whether condition $2$ is $true$ can be tested with certainty in deterministic polynomial time. Thus, we can say that answer to an instance of $A$ is $false$ in polynomial time (in case ... |
|complexity-theory|np-complete|polynomial-time-reductions| |
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To borrow part of a description from a similar but distinct question: there exist 2^(2^N) different functions which accept N binary inputs and return a 1 bit output. For the purposes of my question I am calling these N-input-gates.
I want to know if there exists a general algorithm which can always describe an optim... |
Is there an algorithm to implement N-input-gates using smaller gates? |
|optimization|boolean-algebra|binary|digital-circuits|bit-manipulation| |
null |
I have a function $f$ such that is the sum of big O terms, such as
$$f=\sum_{i=1}^x i +O(x)+O(x-1)+O(x-2)+\ldots$$
where $x$ is a positive integer. Am I correct in that I can simply write
$$f=\sum_{i=1}^x i + O(x)$$
because I just need to write the bigger value for the big O?
Any intuitive, even inform... |
Adding O(x)+O(x-1)+O(x-2)+ |
|asymptotics|big-o-notation| |
There is most likely no polynomial-time algorithm, i.e., no approach that will be efficient on all inputs. The problem is $\Sigma_2^P$-complete, i.e., at least as hard or even harder than NP-hard, even when limiting to a basis of standard two-input gates. However there are heuristic methods.
See also https://en.wi... |
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Having a complete bipartite graph with parts $A$ and $B$, which is edge-weighted, is there a way to compute a maximum flow such that:
1. Only a constant number $n$ of vertices from $A$ are used.
2. An edge points to each of the vertices in $B$.
In simple terms, *pruning* the original graph so that I get $n$ clus... |
Weighted bipartite maximum flow with a fixed number of vertices |
Having a complete bipartite graph with parts $A$ and $B$, which is edge-weighted, is there a way to compute a maximum flow such that:
1. Only a constant number $n$ of vertices from $A$ are used.
2. An edge points to each of the vertices in $B$.
In essence, which nodes from $A$ and which remaining edges to remo... |
Having a complete bipartite graph with parts $A$ and $B$, which is edge-weighted, is there a way to compute a maximum flow such that:
1. Only a constant number $n$ of vertices from $A$ are used.
2. An edge points to each of the vertices in $B$.
In essence, which nodes from $A$ and which remaining edges to remo... |
Having a complete bipartite graph with parts $A$ and $B$, which is edge-weighted, is there a way to compute a subgraph maximizing the sum of all its weights and:
1. Only a constant number $n$ of vertices from $A$ are used.
2. An edge points to each of the vertices in $B$.
In essence, which nodes from $A$ and whi... |
Weighted bipartite maximum cost with a fixed number of vertices |
Having a complete bipartite graph with parts $A$ and $B$, which is edge-weighted, is there a way to compute a subgraph maximizing the sum of all its weights and:
1. Only a constant number $n$ of vertices from $A$ are used.
2. At most one edge points to each of the vertices in $B$.
In essence, which nodes from $A... |
Having a complete bipartite graph with parts $A$ and $B$, which is edge-weighted, is there a way to compute a subgraph with the maximum sum of all weights and:
1. Only a constant number $n$ of vertices from $A$ are used.
2. At most one edge points to each of the vertices in $B$.
In essence, which nodes from $A$ ... |
SCoP, Iteration Domain in Polyhedral Optimization and use of Presburger arithmetic |
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