A special role of Boolean quadratic polytopes among other combinatorial polytopes

TL;DR

This paper classifies various combinatorial polytopes related to NP-complete problems, highlighting the unique role of Boolean quadratic polytopes as faces of many other polytope families.

Contribution

It introduces a method for comparing polytope families via affine reductions and reveals Boolean quadratic polytopes' special position among combinatorial polytopes.

Findings

Boolean quadratic polytopes are faces of many other polytope families.

Two main classes of polytope families are identified based on affine reducibility.

Boolean quadratic polytopes are simpler than polytopes from other complex problems.

Abstract

We consider several families of combinatorial polytopes associated with the following NP-complete problems: maximum cut, Boolean quadratic programming, quadratic linear ordering, quadratic assignment, set partition, set packing, stable set, 3-assignment. For comparing two families of polytopes we use the following method. We say that a family $P$ is affinely reduced to a family $Q$ if for every polytope $p\in P$ there exists $q\in Q$ such that $p$ is affinely equivalent to $q$ or to a face of $q$, where $\dim q = O((\dim p)^k)$ for some constant $k$. Under this comparison the above-mentioned families are splitted into two equivalence classes. We show also that these two classes are simpler (in the above sence) than the families of poytopes of the following problems: set covering, traveling salesman, 0-1 knapsack problem, 3-satisfiability, cubic subgraph, partial ordering. In particular, Boolean quadratic polytopes appear as faces of polytopes in every of the mentioned families.