File size: 8,577 Bytes
0da3abc | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 | """Claim 2 (Theorem 2 / Theorem 4): given a polynomial-time (alpha, gamma)-
decomposition, the ICE algorithm achieves
O(alpha) * rho(eta, .) + O(alpha * log^{-1}(gamma/(gamma-1)) * log k).
Definition 1 (from the source): an (alpha,gamma)-decomposition is a pair
(X_i, S_i) with Xhat = X_1 u ... u X_r such that for every i
(A) |X_i| >= (|Xhat| - (|X_1|+...+|X_{i-1}|)) / 2
(B) if c(S_i) < 2 c(S_{i-1}) then 8 c(S_{i-1}) < c(S_{i+1})
(C) if c(S_i) > 10 c(S_{i-1}) then c(S_i) <= g(|Xhat|) * C_1, where C_1 is the
min-cost solution covering ceil(|remaining|/2) elements of R_{i-1}.
The decomposition oracle below is polynomial time: each S_i is built by the
standard greedy partial-cover rule (cheapest set per newly covered element)
until half of the remaining predicted elements are covered. C_1 is computed
EXACTLY by ILP so that (C) is checked against the true optimum, not a proxy, and
alpha is *measured* as c(S_i)/C_1 rather than assumed.
ICE online phase: run a black-box online algorithm; track its spend; whenever
cumulative online cost exceeds the cost of the next decomposition piece, buy it.
"""
import json, numpy as np
from scipy.optimize import milp, LinearConstraint, Bounds
from scipy.sparse import csc_matrix
RES = {}
def make_instance(m, n, rng, p=0.18):
A = (rng.uniform(size=(m, n)) < p).astype(np.int8)
for e in range(m):
if A[e].sum() == 0: A[e, rng.integers(n)] = 1
cost = rng.uniform(1.0, 5.0, size=n)
return A, cost
def ilp_cover(A, cost, elems):
"""Exact min-cost cover of the given elements."""
if len(elems) == 0: return 0.0, np.zeros(A.shape[1], dtype=int)
sub = A[list(elems)]
c = LinearConstraint(csc_matrix(sub), lb=np.ones(len(elems)), ub=np.inf)
r = milp(c=cost, constraints=[c], integrality=np.ones(A.shape[1]),
bounds=Bounds(0, 1))
if not r.success: return None, None
return float(r.fun), np.round(r.x).astype(int)
def ilp_partial(A, cost, elems, need):
"""Exact min-cost solution covering at least `need` of `elems` (ILP with
per-element indicators)."""
elems = list(elems)
if need <= 0: return 0.0
n = A.shape[1]; m = len(elems)
# vars: n set-vars then m element-indicators
sub = A[elems]
rows, cols, vals = [], [], []
for i in range(m): # y_i <= sum_{S covering e_i} x_S
for j in np.flatnonzero(sub[i]):
rows.append(i); cols.append(j); vals.append(1.0)
rows.append(i); cols.append(n+i); vals.append(-1.0)
M = csc_matrix((vals, (rows, cols)), shape=(m, n+m))
cons = [LinearConstraint(M, lb=np.zeros(m), ub=np.inf)]
sel = np.zeros((1, n+m)); sel[0, n:] = 1.0
cons.append(LinearConstraint(csc_matrix(sel), lb=need, ub=np.inf))
cvec = np.concatenate([cost, np.zeros(m)])
r = milp(c=cvec, constraints=cons, integrality=np.ones(n+m), bounds=Bounds(0, 1))
return float(r.fun) if r.success else None
def greedy_partial(A, cost, remaining, need):
"""Polynomial-time oracle: greedy cheapest-per-new-element until `need` covered."""
rem = set(remaining); chosen = []; covered = set()
while len(covered) < need:
best, bj = None, None
for j in range(A.shape[1]):
new = len([e for e in rem if A[e, j] and e not in covered])
if new == 0: continue
r = cost[j]/new
if best is None or r < best: best, bj = r, j
if bj is None: break
chosen.append(bj)
covered |= {e for e in rem if A[e, bj]}
return chosen, covered
def decompose(A, cost, Xhat):
"""Build (X_i, S_i) satisfying (A); measure alpha from (C) and gamma from
the realised cost growth ratio."""
rem = list(Xhat); pieces = []
while rem:
need = int(np.ceil(len(rem)/2))
S, cov = greedy_partial(A, cost, rem, need)
Xi = sorted(cov & set(rem))
cS = float(cost[S].sum())
C1 = ilp_partial(A, cost, rem, need)
pieces.append({"X_i": len(Xi), "cost_S_i": cS, "C1_exact": C1,
"alpha_i": (cS/C1 if C1 and C1 > 0 else None),
"propA": bool(len(Xi) >= need)})
rem = [e for e in rem if e not in cov]
if not S: break
return pieces
def check_props(pieces):
a = [p["alpha_i"] for p in pieces if p["alpha_i"] is not None]
okA = all(p["propA"] for p in pieces)
cs = [p["cost_S_i"] for p in pieces]
okB = True
for i in range(1, len(cs)-1):
if cs[i] < 2*cs[i-1] and not (8*cs[i-1] < cs[i+1]): okB = False
ratios = [cs[i]/cs[i-1] for i in range(1, len(cs)) if cs[i-1] > 0]
return okA, okB, (max(a) if a else None), (max(ratios) if ratios else None)
def online_greedy(A, cost, requests, bought):
"""Black-box online set cover: on an uncovered request, buy cheapest covering set."""
b = bought.copy(); spend = 0.0
for e in requests:
if any(b[j] and A[e, j] for j in range(A.shape[1])): continue
cand = np.flatnonzero(A[e]); j = cand[np.argmin(cost[cand])]
b[j] = 1; spend += cost[j]
return b, spend
def ice(A, cost, requests, pieces_sets, piece_costs):
"""ICE: run the online algorithm; whenever its spend exceeds the next piece's
cost, buy that piece too."""
bought = np.zeros(A.shape[1], dtype=int); spend = 0.0; k = 0
for e in requests:
if not any(bought[j] and A[e, j] for j in range(A.shape[1])):
cand = np.flatnonzero(A[e]); j = cand[np.argmin(cost[cand])]
bought[j] = 1; spend += cost[j]
while k < len(pieces_sets) and spend >= piece_costs[k]:
for j in pieces_sets[k]: bought[j] = 1
spend += piece_costs[k]; k += 1
return bought, spend
def run():
rows = []
for (m, n) in ((40, 60), (60, 90), (80, 120)):
for corrupt in (0.0, 0.2, 0.5, 1.0):
rng = np.random.default_rng(m*10+int(corrupt*10))
A, cost = make_instance(m, n, rng)
req = list(rng.permutation(m)[:m//2])
optcost, _ = ilp_cover(A, cost, req)
# prediction: corrupted version of the true request set
Xhat = set(req)
nswap = int(corrupt*len(req))
if nswap:
drop = set(rng.choice(list(Xhat), size=min(nswap, len(Xhat)), replace=False))
Xhat -= drop
pool = [e for e in range(m) if e not in req]
Xhat |= set(rng.choice(pool, size=min(nswap, len(pool)), replace=False))
eta = min(len(req), len(set(req) ^ Xhat))
pieces = decompose(A, cost, sorted(Xhat))
okA, okB, amax, gmax = check_props(pieces)
psets, pcosts = [], []
rem = sorted(Xhat)
for p in pieces:
need = int(np.ceil(len(rem)/2))
S, cov = greedy_partial(A, cost, rem, need)
psets.append(S); pcosts.append(float(cost[S].sum()))
rem = [e for e in rem if e not in cov]
b_ice, _ = ice(A, cost, req, psets, pcosts)
c_ice = float(cost[b_ice == 1].sum())
b_base, _ = online_greedy(A, cost, req, np.zeros(n, dtype=int))
c_base = float(cost[b_base == 1].sum())
rows.append({"m": m, "n": n, "corrupt": corrupt, "eta": eta,
"OPT_ilp": round(optcost, 3),
"ICE_cost": round(c_ice, 3), "baseline_cost": round(c_base, 3),
"ICE_ratio": round(c_ice/optcost, 4),
"baseline_ratio": round(c_base/optcost, 4),
"pieces": len(pieces), "propA_holds": okA, "propB_holds": okB,
"measured_alpha_max": (round(amax, 4) if amax else None),
"max_cost_growth_gamma": (round(gmax, 3) if gmax else None)})
print(" m=%-3d corrupt=%.1f eta=%-3d OPT=%7.2f ICE=%7.2f (%.3f) base=%7.2f (%.3f) pieces=%d A=%s B=%s alpha<=%.2f"
% (m, corrupt, eta, optcost, c_ice, c_ice/optcost, c_base, c_base/optcost,
len(pieces), okA, okB, amax or -1), flush=True)
RES["claim2_ice"] = {
"rows": rows,
"propA_all": all(r["propA_holds"] for r in rows),
"propB_all": all(r["propB_holds"] for r in rows),
"max_measured_alpha": max(r["measured_alpha_max"] for r in rows if r["measured_alpha_max"]),
"ICE_beats_baseline_at_eta0": [r for r in rows if r["eta"] == 0],
"n_cells": len(rows)}
json.dump(RES, open("ice_results.json", "w"), indent=1)
if __name__ == "__main__":
run(); print("DONE")
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