The class of algorithms for approximating reasoning tasks presented in this paper is based on approximating the general bucket elimination framework. The algorithms have adjustable levels of accuracy and e ciency, and they can be applied uniformly across many areas and problem tasks. We introduce these algorithms in the context of combinatorial optimization and probabilistic inference. 1 Overview Bucket elimination is a unifying algorithmic framework that generalizes dynamic programming to enable many complex problem-solving and reasoning activities. Amongthe algorithmsthatcan be accommodatedwithin this framework are directional resolution for propositional satis ability Dechter and Rish, 1994 , adaptive consistency for constraint satisfaction Dechter and Pearl, 1987 , Fourier and Gaussian eliminationfor linear inequalities Lassez and Mahler, 1992 , and dynamic programming for combinatorial optimization Bertele and Brioschi, 1972 . Many algorithms for probabilistic inference, such as ...