MAX-CSP, Graph Cuts and Statistical Physics

Baker’s technique, which was created over three decades ago, is a powerful tool for designing polynomial time approximation schemes (PTAS) for NP-hard optimization problems on planar graphs and their generalizations. In this paper, we propose a unified framework to formulate the optimization problems where the local constraints of these problems are encoded by functions attached on the vertices. This framework has much stronger ability for modelling real world problems than classic combinatorial optimization problems. We prove that when the function fi attached on i ∈ V is a liberal function (fi ≥ 0) for all i ∈ V , then there is a PTAS for computing the max-sum of fi on planar graphs. We also prove that computing the min-sum of fi does not admit PTAS even on planar graphs unless P = NP. But if the set of liberal functions satisfies the balance property, we have a PTAS for computing the min-sum. The approximation algorithms for computing max-sum and min-sum imply that the max-product and minproduct can also be well approximated in many cases. These results are further generalized to graphs with bounded local treewidth, H-minor-free graphs, d-dimensional geometric graphs with bounded density and graphs with bounded number of crossings per edge. Our results lead to PTASs for MAX-CUT, MAX-DICUT, MAX-k-CUT on these graphs. We also prove that if the corresponding factor graph of a CNF formula φ can be transformed into these graphs through X − F contractions, then computing the MAX-SAT of φ has a PTAS. This result can be extended to MAX-CSP problems in a similar way. Our technique generalizes Baker’s technique and many existing methods of graph decomposition. Our results make contributions to many important computation problems in various fields such as communication scheduling in wireless networks, task allocation for large-scale distributed database systems, MAP inference on graphical models, energy minimization in statistical physics and many applications in computer vision. State Key Laboratory for Novel Software Technology, Nanjing University, China. Email: [email protected] This work was proposed and finished when I was a full-time research intern at Microsoft Research Asia.