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The theoretical models providing mathematical abstractions for severalsignificant optimization problems in machine learning, combinatorialoptimization, computer vision and statistical physics have intrinsicsimilarities. We propose a unified framework to model these computation taskswhere the structures of these optimization problems are encoded by functionsattached on the vertices and edges of a graph. We show that computing MAX 2-CSPadmits polynomial-time approximation scheme (PTAS) on planar graphs, graphswith bounded local treewidth, $H$-minor-free graphs, geometric graphs withbounded density and graphs embeddable with bounded number of crossings peredge. This implies computing MAX-CUT, MAX-DICUT and MAX $k$-CUT admits PTASs onall these classes of graphs. Our method also gives the first PTAS for computingthe ground state of ferromagnetic Edwards-Anderson model without externalmagnetic field on $d$-dimensional lattice graphs. These results are widelyapplicable in vision, graphics and machine learning.

Approximate MAP Estimation for Pairwise Potentials via Baker's Technique