Bucket Elimination: a Unifying Framework for Structure-driven Inference

Bucket elimination is an algorithmic framework that generalizes dynamic programming to accommodate many complex problem-solving and reasoning tasks. Algorithms such as directional-resolution for proposi-tional satissability, adaptive-consistency for constraint satisfaction, Fourier and Gaussian elimination, for solving linear equalities and inequalities and dynamic programming for combinatorial optimization, can all be accommodated within the bucket elimination framework. Many probabilistic inference tasks can likewise be expressed as bucket-elimination algorithms. These include: belief updating, nding the most probable explanation and expected utility maximization. All these algorithms share the same performance guarantees; all are time and space exponential in the induced-width of the problem's interaction graph. While elimination strategies have extensive demands on memory, pure "conditioning" algorithms require only linear space. Conditioning is a generic name for algorithms that split a problem into subproblems by instantiating a subset of variables, called a conditioning set, or a cut-set. Search algorithms such as backtracking for constraint satisfaction, Davis-Putnam for propositional satissability and branch and bound for combinatorial optimization belong to this class. This paper will demonstrate the use of elimination and conditioning algorithms across areas such as constraint processing, propositional satis-ability, probabilistic inference and decision theoretic planning and proposes ways of combining conditioning with elimination that can be used to trade space for time.