Bucket and Mini-bucket Schemes for M Best Solutions over Graphical Models

The paper focuses on the task of generating the first m best solutions for a combinatorial optimization problem defined over a graphical model (e.g., the m most probable explanations for a Bayesian network). We show that the mbest task can be expressed within the unifying framework of semirings making known inference algorithms defined and their correctness and completeness for the m-best task immediately implied. We subsequently describe elim-m-opt, a new bucket elimination algorithm for solving the m-best task, provide algorithms for its defining combination and marginalization operators and analyze its worstcase performance. An extension of the algorithm to the mini-bucket framework provides bounds for each of the m-best solutions. Empirical demonstrations of the algorithms with emphasis on their potential for approximations are provided.