Methods for advancing combinatorial optimization over graphical models

OF THE DISSERTATION Methods for advancing combinatorial optimization over graphical models By Natalia Flerova Doctor of Philosophy in Computer Science University of California, Irvine, 2015 Rina Dechter, Chair Graphical models are a well-known convenient tool to describe complex interactions between variables. A graphical model defines a function over many variables that factors over an underlying graph structure. One of the popular tasks over graphical models is that of combinatorial optimization. Although many algorithms have been developed with this task in mind, the vast majority are designed to find an optimal solution, minimum or maximum, of an objective function. In many applications, however, it is desirable to obtain not just a single optimal solution, but a set of the first m best solutions for some integer m. The main part of this dissertation focuses on this problem, which we call the m-best optimization task. We show that the m-best 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 worst-case performance. An extension of the algorithm to the mini-bucket framework provides bounds for each of the m best solutions.