The Lasserre Hierarchy in Almost Diagonal Form

TL;DR

This paper introduces a new characterization of the Lasserre hierarchy using almost diagonal moment matrices, enabling the derivation of strong integrality gap bounds for capacitated covering problems.

Contribution

It presents a novel almost diagonal form of the Lasserre hierarchy and a modular method to analyze positive semi-definiteness, leading to new integrality gap lower bounds.

Findings

The integrality gap for min-knapsack remains large at level n-1.

The integrality gap for min-sum of tardy jobs is unbounded at level Ω(√n).

Both problems admit FPTAS despite large gaps.

Abstract

The Lasserre hierarchy is a systematic procedure for constructing a sequence of increasingly tight relaxations that capture the convex formulations used in the best available approximation algorithms for a wide variety of optimization problems. Despite the increasing interest, there are very few techniques for analyzing Lasserre integrality gaps. Satisfying the positive semi-definite requirement is one of the major hurdles to constructing Lasserre gap examples. We present a novel characterization of the Lasserre hierarchy based on moment matrices that differ from diagonal ones by matrices of rank one (almost diagonal form). We provide a modular recipe to obtain positive semi-definite feasibility conditions by iteratively diagonalizing rank one matrices. Using this, we prove strong lower bounds on integrality gaps of Lasserre hierarchy for two basic capacitated covering problems. For the min-knapsack problem, we show that the integrality gap remains arbitrarily large even at level $n-1$ of Lasserre hierarchy. For the min-sum of tardy jobs scheduling problem, we show that the integrality gap is unbounded at level $\Omega(\sqrt{n})$ (even when the objective function is integrated as a constraint). These bounds are interesting on their own, since both problems admit FPTAS.