NDLI: Small extended formulation for knapsack cover inequalities from monotone circuits

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Small extended formulation for knapsack cover inequalities from monotone circuits

Content Provider	ACM Digital Library
Author	Bazzi, Abbas Fiorini, Samuel Svensson, Ola Huang, Sangxia
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 ε > 0, there is a $(1/ε)^{O(1)}$ nO(log n)-size LP relaxation with an integrality gap of at most 2 + ε, 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.
Starting Page	2326
Ending Page	2341
Page Count	16
File Format	PDF
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2017-01-16
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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