Complexity of Discrete Energy Minimization Problems

TL;DR

This paper proves that general discrete energy minimization problems are computationally intractable to approximate within any reasonable ratio, even in simplified cases, and classifies various subclasses into complexity categories.

Contribution

It establishes the inapproximability of general energy minimization problems and organizes related problems into a complexity hierarchy.

Findings

General energy minimization is exp-APX-complete.

Planar energy minimization with three or more labels is exp-APX-complete.

Problems are classified into PO, APX, and exp-APX complexity classes.

Abstract

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.