On 2-Level Polytopes Arising in Combinatorial Settings

Abstract

2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that v(P)*f(P) is upper bounded by d*2^(d+1), for a large collection of families of such polytopes P. Here v(P) (resp. f(P)) is the number of vertices (resp. facets) of P, and d is its dimension. Whether this holds for all 2-level polytopes was asked in [Bohn et al., ESA 2015], and experimental results from [Fiorini et al., ISCO 2016] showed it true up to dimension 7. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures. This leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching polytopes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree.