The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev [32,33] proved that the problem is NP-hard to approximate within a factor 2 − ε, assuming the Unique Games Conjecture (UGC). This is tight because the problem has an easy 2-approximation algorithm. Without resorting to the UGC, the best inapproximability result for the problem is due to Dinur and Safra [17,18]: vertex cover is NP-hard to approximate within a factor 1.3606.We prove the following unconditional result about linear programming (LP) relaxations of the problem: every LP relaxation that approximates vertex cover within a factor 2 − ε has superpolynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations (as well as SDP relaxations) that approximate the independent set problem within any constant factor have super-polynomial size. ContributionWe consider the general model of LP relaxations as in [13], see also [10]. Given an n-vertex graph G = (V, E), a system of linear inequalities Ax b in R d , where d ∈ N is arbitrary, defines an LP relaxation of vertex cover (on G) if the following conditions hold:Linear objective: For every vertex-costs c ∈ R V + , we have an affine function (degree-1 polynomial) Consistency: For all vertex coversFor every vertex-costs c ∈ R V + , the LP min{ f c (x) | Ax b} provides a guess on the minimum cost of a vertex cover. This guess is always a lower bound on the optimum.We allow arbitrary computations for writing down the LP, and do not bound the size of the coefficients. We only care about the following two parameters and their relationship: the size of the LP relaxation, defined as the number of inequalities in Ax b, and the (graph-specific) integrality gap which is the worst-case ratio over all vertex-costs between the true optimum and the guess provided by the LP, for this particular graph G and LP relaxation.This framework subsumes the polyhedral-pair approach in extended formulations [8]; see also [43]. We refer the interested reader to the surveys [15,28] for an introduction to extended formulations; see also Section 4 for more details.In this paper, we prove the following result about LP relaxations of vertex cover and, as a byproduct, independent set. 1 Theorem 1.1. For infinitely many values of n, there exists an n-vertex graph G such that: (i) Every size-n o(log n/ log log n) LP relaxation of vertex cover on G has integrality gap 2 − o(1); (ii) Every sizen o(log n/ log log n) LP relaxation of independent set on G has integrality gap ω(1).This solves an open problem that was posed both by Singh [51] and Chan, Lee, Raghavendra and Steurer [13]. In fact, Singh conjectured that every compact (that is, polynomial size), symmetric extended formulation for vertex cover has integrality gap at least 2−ε. We prove that his conjecture holds, even if asymmetric extended formulations are allowed. 2 Our result for the independent set problem is even stronger than Theorem 1.1, as we are also able to rul...
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) n O(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.
Abstract. We show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. Assuming a natural but nontrivial generalisation of the bipartite structural hardness result of [1], we obtain a hardness of 2 − for the problem of minimising the makespan for scheduling precedence-constrained jobs with preemption on identical parallel machines. This matches the best approximation guarantee for this problem [6,4]. Assuming the same hypothesis, we also obtain a super constant inapproximability result for the problem of scheduling precedence-constrained jobs on related parallel machines, making progress towards settling an open question in both lists of ten open questions by Williamson and Shmoys [17], and by Schuurman and Woeginger [14]. The study of structural hardness of k-partite graphs is of independent interest, as it captures the intrinsic hardness for a large family of scheduling problems. Other than the ones already mentioned, this generalisation also implies tight inapproximability to the problem of minimising the weighted completion time for precedence-constrained jobs on a single machine, and the problem of minimising the makespan of precedenceconstrained jobs on identical parallel machine, and hence unifying the results of Bansal and Khot [1] and Svensson [15], respectively.
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