1 research outputs found
High Degree Sum of Squares Proofs, Bienstock-Zuckerberg hierarchy and Chvatal-Gomory cuts
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.Comment: Revised version with some small typos correcte