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Volume 14 (2018) Article 14 pp. 1-29
Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits
Received: December 9, 2016
Revised: August 13, 2018
Published: December 2, 2018
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Keywords: extended formulations, communication complexity, linear programming, knapsack
ACM Classification: G.1.6
AMS Classification: 68Q25, 90C59

Abstract: [Plain Text Version]

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. Previously, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our techniques are also sufficiently versatile to give analogous results for the closely related flow cover inequalities that are used to strengthen relaxations for scheduling and facility location problems.

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.

A conference version of this paper appeared in the Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA 2017).