On 2-level polytopes arising in combinatorial settings

TL;DR

This paper investigates 2-level polytopes in combinatorial contexts, establishing an upper bound on the product of vertices and facets for many families, and explores their structural properties and relations to combinatorial objects.

Contribution

It proves an upper bound on v(P)*f(P) for many 2-level polytopes, addressing a question from prior research, and provides new insights into their combinatorial structures.

Findings

Proved v(P)*f(P) ≤ d*2^(d+1) for many 2-level polytopes.

Derived a trade-off formula for the number of cliques and stable sets in graphs.

Described stable matching polytopes as affine projections of order polytopes.

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.