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Revision #2 to TR17-157 | 17th September 2018 12:22

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

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Revision #2
Authors: Monaldo Mastrolilli
Accepted on: 17th September 2018 12:22
Downloads: 662
Keywords: 


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 constructible 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) 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.



Changes to previous version:

Revised version with additional and better explanations.


Revision #1 to TR17-157 | 24th November 2017 16:34

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





Revision #1
Authors: Monaldo Mastrolilli
Accepted on: 24th November 2017 16:34
Downloads: 856
Keywords: 


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 constructible 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.



Changes to previous version:

Added a comment on a recent related claim by Fiorini et al.


Paper:

TR17-157 | 13th October 2017 00:45

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


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 constructible 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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