PTAS for MAP Assignment on Pairwise Markov Random Fields in Planar Graphs

TL;DR

This paper introduces a Polynomial-Time Approximation Scheme (PTAS) for the MAP assignment problem on Pairwise Markov Random Fields with non-negative weights in planar graphs, providing a practical approach with theoretical guarantees.

Contribution

It presents the first PTAS for MAP assignment on Pairwise Markov Random Fields in planar graphs, and shows hardness results for negative weights, extending to correlation clustering.

Findings

PTAS achieves near-optimal solutions efficiently.

Hardness results show no approximation for negative weights unless P=NP.

PTAS is applicable to image processing tasks.

Abstract

We present a PTAS for computing the maximum a posteriori assignment on Pairwise Markov Random Fields with non-negative weights in planar graphs. This algorithm is practical and not far behind state-of-the-art techniques in image processing. MAP on Pairwise Markov Random Fields with (possibly) negative weights cannot be approximated unless P = NP, even on planar graphs. We also show via reduction that this yields a PTAS for one scoring function of Correlation Clustering in planar graphs.