High Degree Sum of Squares Proofs, Bienstock-Zuckerberg hierarchy and Chvatal-Gomory cuts

TL;DR

This paper introduces a novel SOS hierarchy that captures the Bienstock-Zuckerberg hierarchy and efficiently approximates certain combinatorial polytopes, surpassing standard low-degree SOS methods.

Contribution

It presents a new polynomial-time SOS hierarchy based on high-degree polynomials that reproduces the Bienstock-Zuckerberg hierarchy and improves approximation of CG cuts.

Findings

Reproduces Bienstock-Zuckerberg hierarchy as a polynomial-sized LP.

Optimizes over polytopes from constant rounds of CG-cuts with small error.

Uses a novel basis of high-degree polynomials for SOS hierarchies.

Abstract

Chvatal-Gomory (CG) cuts and the Bienstock-Zuckerberg hierarchy capture useful linear programs that the standard bounded degree Lasserre/Sum-of-Squares SOS hierarchy fails to capture. In this paper we present a novel polynomial time SOS hierarchy for 0/1 problems with a custom subspace of high degree polynomials (not the standard subspace of low-degree polynomials). We show that the new SOS hierarchy recovers the Bienstock-Zuckerberg hierarchy. Our result implies a linear program that reproduces the Bienstock-Zuckerberg hierarchy as a polynomial sized, efficiently constructive extended formulation that satisfies all constant pitch inequalities. The construction is also very simple, and it is fully defined by giving the supporting polynomials. Moreover, for a class of polytopes (e.g. set covering and packing problems), the resulting SOS hierarchy optimizes in polynomial time over the polytope resulting from any constant rounds of CG-cuts, up to an arbitrarily small error. Arguably, this is the first example where different basis functions can be useful in asymmetric situations to obtain a hierarchy of relaxations.