Revisiting MAP Estimation, Message Passing and Perfect Graphs

Given a graphical model, one of the most useful queries is to find the most likely configuration of its variables. This task, known as the maximum a posteriori (MAP) problem, can be solved efficiently via message passing techniques when the graph is a tree, but is NPhard for general graphs. Jebara (2009) shows that the MAP problem can be converted into the stable set problem, which can be solved in polynomial time for a broad class of graphs known as perfect graphs via a linear programming relaxation technique. This is a result of great theoretical interest. However, the article additionally claims that max-product linear programming (MPLP) message passing techniques of Globerson and Jaakkola (2007) are also guaranteed to solve these problems exactly and efficiently. We investigate this claim, show that it does not hold, and repair it with alternative message passing algorithms.

[1]  Vladimir Kolmogorov,et al.  Convergent Tree-Reweighted Message Passing for Energy Minimization , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[2]  Martin J. Wainwright,et al.  On the Optimality of Tree-reweighted Max-product Message-passing , 2005, UAI.

[3]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[4]  Tony Jebara,et al.  MAP Estimation, Message Passing, and Perfect Graphs , 2009, UAI.

[5]  P. Seymour,et al.  The Strong Perfect Graph Theorem , 2002, math/0212070.

[6]  Nikos Komodakis,et al.  Beyond Loose LP-Relaxations: Optimizing MRFs by Repairing Cycles , 2008, ECCV.

[7]  Devavrat Shah,et al.  Maximum weight matching via max-product belief propagation , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[8]  L. Lovász.  Normal Hypergraphs and the Weak Perfect Graph Conjecture , 1984 .

[9]  Solomon Eyal Shimony,et al.  Finding MAPs for Belief Networks is NP-Hard , 1994, Artif. Intell..

[10]  N. Biggs.  GEOMETRIC ALGORITHMS AND COMBINATORIAL OPTIMIZATION: (Algorithms and Combinatorics 2) , 1990 .

[11]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .

[12]  Paul D. Seymour,et al.  Recognizing Berge Graphs , 2005, Comb..

[13]  Martin J. Wainwright,et al.  MAP estimation via agreement on trees: message-passing and linear programming , 2005, IEEE Transactions on Information Theory.

[14]  Tommi S. Jaakkola,et al.  Tree Block Coordinate Descent for MAP in Graphical Models , 2009, AISTATS.

[15]  X. Jin.  Factor graphs and the Sum-Product Algorithm , 2002 .

[16]  Tommi S. Jaakkola,et al.  Tightening LP Relaxations for MAP using Message Passing , 2008, UAI.

[17]  László Lovász,et al.  Normal hypergraphs and the perfect graph conjecture , 1972, Discret. Math..

[18]  Tommi S. Jaakkola,et al.  Fixing Max-Product: Convergent Message Passing Algorithms for MAP LP-Relaxations , 2007, NIPS.

[19]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .