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Counting the Optimal Solutions in Graphical Models Rina Dechter Radu Marinescu University of California, Irvine IBM Research Motivation and Contribution Combinatorial optimization in graphical models Solution that optimizes a global

  1. Counting the Optimal Solutions in Graphical Models Rina Dechter Radu Marinescu University of California, Irvine IBM Research

  2. Motivation and Contribution ● Combinatorial optimization in graphical models – Solution that optimizes a global objective function ● NP-hard : exponentially many terms ● #opt: count the optimal solutions – Naive brute-force approaches based on enumeration ● Infeasible in practice if many optimal solutions – Introduce efficient variable elimination and search based algorithms that do not rely on enumeration

  3. #opt ● Formally: – Computed efficiently, without enumeration value ● The #opt semiring: count - combination operator - marginalization operator Property: distributes over

  4. Exact Algorithms for #opt ● Variable Elimination (VE) – Eliminate variables following an ordering – Local computations facilitated by the distributivity property of the semiring – Complexity : O(n exp(w*)) w* - treewidth ● AND/OR Branch-and-Bound Search (AOBB) – Explore the context-minimal AND/OR search graph – Heuristic evaluation function to prune unpromising regions of the search space – Complexity : O(n exp(w*)) w* - treewidth ● In practice, more efficient due to pruning

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