Supermodular Decomposition of Structural Labeling Problem

We consider the problem of selecting labels in vertices of the graph in order to maximize sum of weights, assigned to pairs of labels ( (max,+) problem, MAP MRF, minimization of Gibbs energy). We construct upper bounds on the goal function as sum of two auxiliary superand submodular problems. The tightest possible bound of this type can be found by solving a special case of LP. It's solution allows to fix some part of optimal configuration of the initial problem. Introduction. In the pattern recognition we know fundamental problems, which arise as estimation of maximal a posteriori probability configuration of Markov Random Field, as search of the optimal labeling, as minimization of Gibbs energy. Optimization models of this type play an important role in other areas: in statistical mechanics they arise as minimization of Hamiltonian energy of spin glass systems, in communication theory as problems of optimal error-correcting decoding. Their abstract formulations are studied in operations research, discrete mathematics, combinatorial theory. Problem formulation. Let = ( , ) G T E be a directed graph, where ( ) 2 T E ⊆ . Elements of set T we’ll refer as objects, and elements of E as pairs. United set = T E I ∪ we’ll call as structure of the problem. Let K be finite set of labels. Let us call mapping : k T K → a labeling ( T k K ∈ ). By : k K τ τ → ( k K τ τ ∈ ) we’ll denote restriction of labeling k to the subset τ ∈ I . Let ;k τ τ θ ∈R be 1 This paper was supported by the INTAS project PRINCESS 04-77-7347. source: Shekhovtsov, A. “Supermodular decomposition of structural labeling problem”, Control Systems and Machienes, Kiev 2005, accepted, in Russian. http://cmp.felk.cvut.cz/~shekhovt/publications/shekhovtsov-SM-05-art-en.pdf