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AND/OR Branch-and-Bound search for combinatorial optimization in graphical models

Radu Marinescu, Rina Dechter

2009
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Artificial Intelligence
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This is the first of two papers presenting and evaluating the power of a new framework for combinatorial optimization in graphical models, based on AND/OR search spaces. We introduce a new generation of depth-first Branch-and-Bound algorithms that explore the AND/OR search tree using static and dynamic variable orderings. The virtue of the AND/OR representation of the search space is that its size may be far smaller than that of a traditional OR representation, which can translate into
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... nt time savings for search algorithms. The focus of this paper is on linear space search which explores the AND/OR search tree. In the second paper we explore memory intensive AND/OR search algorithms. In conjunction with the AND/OR search space we investigate the power of the mini-bucket heuristics in both static and dynamic setups. We focus on two most common optimization problems in graphical models: finding the Most Probable Explanation in Bayesian networks and solving Weighted CSPs. In extensive empirical evaluations we demonstrate that the new AND/OR Branch-and-Bound approach improves considerably over the traditional OR search strategy and show how various variable ordering schemes impact the performance of the AND/OR search scheme. Definition 1 (directed, undirected graphs). A directed graph is defined by a pair G = {V, E}, where V = {X 1 , . . . , X n } is a set of vertices (nodes), and E = {(X i , X j ) | X i , X j ∈ V } is a set of edges (arcs). If (X i , X j ) ∈ E, we say that X i points to X j . The degree of a vertex is the number of incident arcs to it. For each vertex X i , pa( X i ) or pa i , is the set of vertices pointing to X i in G, while the set of child vertices of X i , denoted ch( X i ), comprises the variables that X i points to. The family of X i , denoted F i , includes X i and its parent vertices. A directed graph is acyclic if it has no directed cycles. An undirected graph is defined similarly to a directed graph, but there is no directionality associated with the edges. Definition 2 (induced width). An ordered graph is a pair (G, d) where G is an undirected graph, and d = X 1 , . . . , X n is an ordering of the nodes. The width of a node is the number of the node's neighbors that precede it in the ordering. The width of an ordering d is the maximum width over all nodes. The induced width of an ordered graph, denoted by w * (d), is the width of the induced ordered graph obtained as follows: nodes are processed from last to first; when node X i is processed, all its preceding neighbors are connected. The induced width of a graph, denoted by w * , is the minimal induced width over all its orderings.

doi:10.1016/j.artint.2009.07.003
fatcat:ygqm7xz4kjavpoquq4ja4mzyeq