Strong Reductions for Extended Formulations

Abstract

We generalize the reduction mechanism for linear programming problems and semidefinite programming problems from [BPZ15] in two ways (1) relaxing the requirement of affineness, and (2) extending to fractional optimization problems.As applications we provide several new LP-hardness and SDP-hardness results, e.g., for the SparsestCut problem, the BalancedSeparator problem, the MaxCut problem and the Matching problem on 3-regular graphs. We also provide a new, very strong Lasserre integrality gap for the IndependentSet problem, which is strictly greater than the best known LP approximation, showing that the Lasserre hierarchy does not always provide the tightest SDP relaxation.

Limitations of Semidefinite Programs for Separable States and Entangled Games

“…Intuitively, LRS can only show that some problem of size n is hard if one can simultaneously embed a hard instance of size m < n into each size-m subset of the size-n input. This is the case for CSP problems as well as for the problems considered in [BPZ15,BPR16].…”

“…Our crucial observation is that our reductions inspired by quantum protocols actually allow one to embed hard problems over {0, 1} n into larger, even infinite-dimensional, domains. Moreover, our reductions are naturally low-degree, which could be a technically interesting point comparing to the affine reductions in [BPZ15,BPR16]. As a result, we can avoid extending [LRS15]'s analysis to each setting, while still extending its results on {0, 1} n to much larger domains, albeit with some loss in parameters.…”

“…On top of that, most reductions used in deriving our SoS hardness are low-degree rather than affine (as in [BPZ15,BPR16]). Finally, the particular class of relaxations that [LRS15] applies to, here referred to as SDP extended formulations, must satisfy certain rather rigid technical constraints, in particular one which we formulate as the embedding property.…”

Affine reductions for LPs and SDPs

“…The conditional SDP inapproximability factors are formulated under the assumption that the Goemans-Williams SDP for MaxCUT is optimal (see Conjecture 4.6), which is compatible with the Unique Games Conjecture. Alternatively, these reductions can be also combined with the 15/16 + ε SDP-hardness of approximation for MaxCUT that was recently established in Braun et al [2015c] to obtain unconditional (but weaker) inapproximability factors. To obtain the respective factors it suffices to replace c GW with 15/16 in the arguments.…”

“…Finally, we would also like to note that our model and reductions have been already applied to obtain inapproximability results for approximate GraphIsomorphism in Braun et al [2015b], exponential lower bound for symmetric SDPs for Matching (see [Braun et al, 2015a, Theorem 3.1]), and(2 − ε)-inapproximability for VertexCover together with inapproximability of IndependentSet within any constant factor in Bazzi et al [2015] improving our Theorem 5.3 by a significantly more involved argument. An extension of this reduction mechanism is presented in Braun et al [2015c] relaxing some of our conditions, which enables the study of fractional optimization problems (such as e.g., Sparsest Cut).…”