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

Monaldo Mastrolilli

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arXiv · 1709.07966

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

Monaldo Mastrolilli

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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.

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Monaldo Mastrolilli. 2017-09-22. High Degree Sum of Squares Proofs, Bienstock-Zuckerberg hierarchy and Chvatal-Gomory cuts. https://arxiv.org/abs/1709.07966

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