Abstract
Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be stated formally as follows: for any $\varepsilon >0$, there is a $(1/\varepsilon)^{O(1)}n^{O(\log n)}$-size LP relaxation with an integrality gap of at most $2+\varepsilon$, where $n$ is the number of items. Prior to this work, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our construction is inspired by a connection between extended formulations and monotone circuit complexity via Karchmer-Wigderson games. In particular, our LP is based on $O(\log^2 n)$-depth monotone circuits with fan-in~$2$ for evaluating weighted threshold functions with $n$ inputs, as constructed by Beimel and Weinreb. We believe that a further understanding of this connection may lead to more positive results complementing the numerous lower bounds recently proved for extended formulations.
Highlights
Capacitated covering problems1 play a central role in combinatorial optimization
We show that the linear programming (LP) relaxation can be constructed in quasi-polynomial time whenever the data is integer and quasipolynomially bounded
We show how to construct in quasi-polynomial time a quasi-polynomial-size LP formulation for min-knapsack with integrality gap at most (2 + ε)
Summary
Capacitated covering problems play a central role in combinatorial optimization. These are the problems modeled by Integer Programs (IPs) of the form n min ∑ cixi Ax b, x ∈ {0, 1}n , i=1 where A is a size-m × n nonnegative matrix and b and c are size-m and size-n nonnegative vectors, respectively. Sn ∈ R+ and demand D ∈ R+, there exists a size-(1/ε)O(1)nO(logn) extended formulation defining an LP relaxation of min-knapsack with integrality gap at most 2 + ε. In this case, using techniques similar to those developed for polynomial-time approximation schemes, they obtained polynomial-size relaxations with integrality gap at most 1 + ε, for any fixed ε > 0 This is, a very different setting and, as the developed inequalities depend on the objective function, they do not generalize to other problems. As a motivational example for the flow cover inequalities, we illustrate in Section 4 how this strengthening works for the Single Demand Facility Location problem, reducing the integrality gap to 2 This approach shares the same drawback that was identified in the case of min-knapsack, which is that the size of the resulting LP formulation becomes exponential.
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