Complexity of Discrete Energy Minimization Problems

Mengtian Li*, Alexander Shekhovtsov, Daniel Huber

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review

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.
Original languageEnglish
Title of host publicationEuropean Conference on Computer Vision - ECCV 2016
Pages834-852
Number of pages19
DOIs
Publication statusPublished - 2016
Event14th European Conference on Computer Vision: ECCV 2016 - Amsterdam, Netherlands
Duration: 8 Oct 201616 Oct 2016

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9906
ISSN (Electronic)0302-9743

Conference

Conference14th European Conference on Computer Vision
Country/TerritoryNetherlands
CityAmsterdam
Period8/10/1616/10/16

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