Higher Order Maximum Persistency and Comparison Theorems

TL;DR

This paper introduces a unified framework for identifying persistent variables in combinatorial optimization problems, improving upon existing methods by providing a sufficient condition and a linear program to find maximal persistent sets.

Contribution

It develops a general theoretical framework for higher order maximum persistency, unifying and extending existing methods through a new linear programming approach.

Findings

The framework explains various existing persistency methods.

The linear program identifies maximal persistent variables.

The approach improves as relaxations are tightened.

Abstract

We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1 polynomial programming). For polyhedral relaxations of such problems it is generally not true that variables integer in the relaxed solution will retain the same values in the optimal discrete solution. Those which do are called persistent. Such persistent variables define a part of a globally optimal solution. Once identified, they can be excluded from the problem, reducing its size. To any polyhedral relaxation we associate a sufficient condition proving persistency of a subset of variables. We set up a specially constructed linear program which determines the set of persistent variables maximal with respect to the relaxation. The condition improves as the relaxation is tightened and possesses all its invariances. The proposed framework explains a variety of existing methods originating from different areas of research and based on different principles. A theoretical comparison is established that relates these methods to the standard linear relaxation and proves that the proposed technique identifies same or larger set of persistent variables.