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Feb 10, 2024 •169 likes •225 views

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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