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We provide a numerical refutation of the developments of Fiorini et al. (2015) * for modelswith disjoint sets of descriptive variables. We also provide an insight into the meaning of the existence of a one-to-one linear map between solutions of such models.

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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.

Motivated by [12], we provide a framework for studying the size of linear programming formulations as well as semidefinite programming formulations of combinatorial optimization problems without encoding them first as linear programs. This is done via a factorization theorem for the optimization problem itself (and not a specific encoding of such). As a result we define a consistent reduction mechanism that degrades approximation factors in a controlled fashion and which, at the same time, is compatible with approximate linear and semidefinite programming formulations. Moreover, our reduction mechanism is a minor restriction of classical reductions establishing inapproximability in the context of PCP theorems. As a consequence we establish strong linear programming inapproximability (for LPs with a polynomial number of constraints) for several problems that are not 0/1-CSPs: we obtain a 3 2 − ε inapproximability for VertexCover (which is not of the CSP type) answering an open question in [12], we answer a weak version of our sparse graph conjecture posed in [6] showing an inapproximability factor of 1 2 +ε for bounded degree IndependentSet, and we establish inapproximability of Max-MULTI-k-CUT (a non-binary CSP). In the case of SDPs, we obtain relative inapproximability results for these problems.

In Rothvoß (Math Program 142(1-2):255-268, 2013) it was shown that there exists a 0/1 polytope (a polytope whose vertices are in {0, 1} n ) such that any higher-dimensional polytope projecting to it must have 2 Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high positive semidefinite extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2 Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations.

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...