Approximate MAP Estimation for Pairwise Potentials via Baker's Technique

TL;DR

This paper introduces a unified framework for approximating complex optimization problems on various graph classes, providing polynomial-time approximation schemes for problems like MAX-CUT and the ground state of certain physical models.

Contribution

It presents the first PTAS for several important problems on specific graph classes using Baker's technique, unifying and extending approximation results across multiple domains.

Findings

PTAS for MAX 2-CSP on planar and bounded treewidth graphs

PTAS for MAX-CUT, MAX-DICUT, and MAX k-CUT on various graph classes

First PTAS for ferromagnetic Edwards-Anderson model on lattice graphs

Abstract

The theoretical models providing mathematical abstractions for several significant optimization problems in machine learning, combinatorial optimization, computer vision and statistical physics have intrinsic similarities. We propose a unified framework to model these computation tasks where the structures of these optimization problems are encoded by functions attached on the vertices and edges of a graph. We show that computing MAX 2-CSP admits polynomial-time approximation scheme (PTAS) on planar graphs, graphs with bounded local treewidth, $H$-minor-free graphs, geometric graphs with bounded density and graphs embeddable with bounded number of crossings per edge. This implies computing MAX-CUT, MAX-DICUT and MAX $k$-CUT admits PTASs on all these classes of graphs. Our method also gives the first PTAS for computing the ground state of ferromagnetic Edwards-Anderson model without external magnetic field on $d$-dimensional lattice graphs. These results are widely applicable in vision, graphics and machine learning.