Integrality Gaps for Colorful Matchings

TL;DR

This paper investigates the integrality gaps of linear programming relaxations for the Bounded Color Matching problem, analyzing how certain graph structures and the Sherali-Adams hierarchy influence the bounds.

Contribution

It establishes lower bounds on integrality gaps, examines the Sherali-Adams hierarchy's behavior, and shows how excluding specific sub-structures improves the relaxation's performance.

Findings

Lower bounds on integrality gaps for various instances.

Analysis of Sherali-Adams hierarchy's effectiveness.

Exclusion of simple sub-structures improves LP relaxation.

Abstract

We study the integrality gap of the natural linear programming relaxation for the \textit{Bounded Color Matching} (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali-Adams "lift-and-project" technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of F\"{u}redi [\emph{Combinatorica}, 1(2):155-162, 1981]. We further leverage this to show upper bounds on the performance of the Sherali-Adams hierarchy when applied to the natural LP relaxation of the BCM problem.