On the Hardest Problem Formulations for the 0/1 Lasserre Hierarchy

TL;DR

This paper characterizes the hardest 0/1 integer and polynomial optimization problems for the Lasserre hierarchy, identifying cases where the hierarchy fails to close the integrality gap even at high levels.

Contribution

It provides a characterization of the most challenging 0/1 problems for the Lasserre hierarchy, revealing limitations in the hierarchy's ability to solve certain problems.

Findings

Identifies problems with integrality gaps at level n-1

Characterizes the set of hardest 0/1 problems for Lasserre hierarchy

Highlights limitations of the hierarchy in certain cases

Abstract

The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the convex relaxations used in the best available approximation algorithms for a wide variety of optimization problems. In this paper we characterize the set of 0/1 integer linear problems and unconstrained 0/1 polynomial optimization problems that can still have an integrality gap at level n-1. These problems are the hardest for the Lasserre hierarchy in this sense.