High Degree Sum of Squares Proofs, Bienstock-Zuckerberg hierarchy and Chvatal-Gomory cuts
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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 (one paragraph). Moreover, for a class of polytopes (e.g. set covering and packing problems) it optimizes, up to an arbitrarily small error, over the polytope resulting from any constant rounds of CG cuts. 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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