A Semidefinite Approach to the $K_i$ Cover Problem

TL;DR

This paper introduces a semidefinite programming approach using theta body relaxations to address the $K_i$-cover problem, achieving polynomial solutions for specific graph classes and analyzing the relaxation's limitations.

Contribution

It presents a novel semidefinite relaxation method for the $K_i$-cover problem, including polynomial-time solutions for certain graphs and insights into the relaxation's convergence and integrality gaps.

Findings

Polynomial-time solvability for certain graph classes.

Theta body relaxations do not converge for $K_n$ with $(n-2)/4$ steps.

An integrality gap of 2 for the second theta body across all graphs.

Abstract

We apply theta body relaxations to the $K_i$-cover problem and show polynomial time solvability for certain classes of graphs. In particular, we give an effective relaxation where all $K_i$-$p$-hole facets are valid, and study its relation to an open question of Conforti et al. For the triangle free problem, we show for $K_n$ that the theta body relaxations do not converge by $(n-2)/4$ steps; we also prove for all $G$ an integrality gap of 2 for the second theta body.