Lower bounds for cube-ideal set-systems

TL;DR

This paper establishes exponential lower bounds on the size of cube-ideal set-systems and linear bounds on their VC dimension, with applications to various problems in graph theory and combinatorial optimization.

Contribution

It provides the first exponential lower bounds for cube-ideal set-systems and connects these bounds to important combinatorial optimization problems.

Findings

Exponential lower bounds on the size of cube-ideal set-systems.

Linear lower bounds on the VC dimension of cube-ideal set-systems.

Applications to strong orientations, perfect matchings, dijoins, and ideal clutters.

Abstract

A set-system $S\subseteq \{0,1\}^n$ is cube-ideal if its convex hull can be described by capacity and generalized set covering inequalities. In this paper, we use combinatorics, convex geometry, and polyhedral theory to give exponential lower bounds on the size of cube-ideal set-systems, and linear lower bounds on their VC dimension. We then provide applications to graph theory and combinatorial optimization, specifically to strong orientations, perfect matchings, dijoins, and ideal clutters, including the Lov\'{a}sz-Plummer conjecture.