On Partial Opimality by Auxiliary Submodular Problems

TL;DR

This paper establishes theoretical relations between three energy minimization techniques, introducing LP-autarky and showing how Kovtun's auxiliary submodular problems relate to LP relaxation and expansion moves, with implications for optimal partial assignments.

Contribution

The paper introduces LP-autarky as a new sufficient condition for optimal partial assignment and links Kovtun's auxiliary submodular problems to LP relaxation, clarifying their limitations and relations.

Findings

Kovtun's auxiliary submodular problems fulfill LP-autarky, a new sufficient condition.

LP relaxation provides optimal partial assignment (persistency) in two-label problems, dominating IK.

Expansion move fixed points coincide with IK's restricted problem under certain truncation rules.

Abstract

In this work, we prove several relations between three different energy minimization techniques. A recently proposed methods for determining a provably optimal partial assignment of variables by Ivan Kovtun (IK), the linear programming relaxation approach (LP) and the popular expansion move algorithm by Yuri Boykov. We propose a novel sufficient condition of optimal partial assignment, which is based on LP relaxation and called LP-autarky. We show that methods of Kovtun, which build auxiliary submodular problems, fulfill this sufficient condition. The following link is thus established: LP relaxation cannot be tightened by IK. For non-submodular problems this is a non-trivial result. In the case of two labels, LP relaxation provides optimal partial assignment, known as persistency, which, as we show, dominates IK. Relating IK with expansion move, we show that the set of fixed points of expansion move with any "truncation" rule for the initial problem and the problem restricted by one-vs-all method of IK would coincide -- i.e. expansion move cannot be improved by this method. In the case of two labels, expansion move with a particular truncation rule coincide with one-vs-all method.