Approximation Limits of Linear Programs (Beyond Hierarchies)

Abstract

We develop a framework for proving approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n 1/2−ǫ )-approximations for CLIQUE require linear programs of size 2 n Ω(ǫ) . This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by linear programs.Our main technical ingredient is a quantitative improvement of Razborov's rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.

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“…The reason the proof due to [6] only works up to n 1/2−δ -approximation (rather than n 1−δ -approximation) is similar to the reason why majority-amplification yields…”

“…We say M is an ε-UDISJ matrix (where ε is to be thought of as a function of n, and UDISJ stands for UNIQUE-SET-DISJOINTNESS) iff it is 2 n × 2 n with rows and columns indexed by subsets x, y ⊆ [n], each entry with |x ∩ y| = 0 is 1, each entry with |x ∩ y| = 1 is 1 − ε, and all other entries are nonnegative. The authors of [6] proved the following two things.…”

“…The paper [6] proved (i) by developing a generalization of the corruption lemma from [35]. Both of the subsequent papers [8,7] employed information-theoretic methods.…”

“…Although communication complexity lower bounds do not seem to imply extended formulation lower bounds in a black-box way, the tools for the former have generally been useful for the latter. This suggests that perhaps an idea analogous to our and-amplification-based communication lower bound proof could be used to "bootstrap" the result of [6] to get the tight lower bound for maximum-clique. In Section 5 we confirm that this is indeed the case, thus obtaining a proof of the tight bound that is simpler than the ones given in [8,7] and avoids their use of information-theoretic methods (instead relying only on the corruption-based argument of [6]).…”

“…The authors of [6] proved that for every positive constant δ < 1/2, so-called n 1/2−δ -approximate extended formulations for a certain natural linear programming encoding of the maximum-clique problem for graphs have complexity 2 Ω(n 2δ ) . Their proof involves developing a generalization of the corruption lemma from [35].…”