Inapproximability of Combinatorial Problems via Small LPs and SDPs

Abstract

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.

Average case polyhedral complexity of the maximum stable set problem

“…The motivation for the streamlined model is to give a direct natural approach to understanding polyhedral complexity, in particular in the context of approximate formulations. Our model is in line with [2,6,8,17].…”

Approximate Constraint Satisfaction Requires Large LP Relaxations

“…One might hope to extend these connections to other types of problems like TSP and finding maximum-weight perfect matchings in general graphs [Rot14,Yan91] or approximations for vertex cover. See [BPZ15] for progress on the latter problem.…”

“…exponential lower bounds on the semidefinite extension complexity of explicit polytopes (like the TSP polytopes). Finally, our models for approximation via linear programs are extended and refined in the work [BPZ15]; the authors show that a suitable notion of reduction within the model allows one to derive lower bounds for additional problems (other than CSPs). LP and SDP hierarchies.…”

Limitations of Semidefinite Programs for Separable States and Entangled Games

“…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.…”