Average case polyhedral complexity of the maximum stable set problem

Abstract

We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend on the input graph, which should be encoded solely in the objective function. There we prove a $2^{\Omega(n/ \log n)}$ lower bound with probability at least $1 - 2^{-2^n}$ for every LP that is exact for a randomly selected set of instances; each graph on at most n vertices being selected independently with probability $p \geq 2^{-\binom{n/4}{2}+n}$. In the non-uniform model, the constraints of the LP may depend on the input graph, but we allow weights on the vertices. The input graph is sampled according to the G(n, p) model. There we obtain upper and lower bounds holding with high probability for various ranges of p. We obtain a super-polynomial lower bound all the way from $p = \Omega(\log^{6+\varepsilon} / n)$ to $p = o (1 / \log n)$. Our upper bound is close to this as there is only an essentially quadratic gap in the exponent, which currently also exists in the worst-case model. Finally, we state a conjecture that would close this gap, both in the average-case and worst-case models

Inapproximability of Combinatorial Problems via Small LPs and SDPs

“…The linear program is constructed from scratch using a matrix factorization. This also extends [7], by allowing affine functions, and using a modification of nonnegative rank, to show that formulation complexity depends only on the slack matrix. Let P * = (P, F * ) be an approximation problem with an optimization problem P = (S, F).…”

“…In particular, we answer an open question regarding the inapproximability of VertexCover (see [12]) and we answer a weak version of our sparse graph conjecture posed in [6].…”

“…For minimization problem, f * := min f in the exact case, and f * = (min f )/ρ for an approximation factor ρ ≥ 1 provided f is nonnegative. This model of outer approximation streamlines the models in [12,7,33], and also captures, simplifies, and generalizes approximate extended formulations from [8,9]; see Section 2.1 for a discussion. A wide class of examples consist of constraint satisfaction problems (CSPs), which we recall now.…”

“…A reduction mechanism for exact extended formulations has been considered in [3,36], where it already provided lower bounds on the exact extension complexity of various polytopes. Our generalization of (encoding dependent) extended formulations to encoding independent formulation complexity is closely related to [12] and [7]. The former studied an encoding-independent model for uniform formulations of CSPs, whereas the latter used a restricted LP version of the model presented here.…”

“…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.…”