A Branch-and-Bound Algorithm for a Family of Pseudo-Boolean Optimization Problems

Let Ω ⊂ {0, 1}n be a set of binary vectors, and let us associate to each vector ω ∈ Ω a real weight. Consider now the problem of finding a conjunction C such that the sum of the weights of the vectors in Ω satisfying C is maximized. This problem can be formulated as a pseudo-Boolean optimization problem in which every term has degree n. This problem is a generalization of the so-called “maximum pattern problem” and has a natural application in an iterative algorithm for regression with binary predictors. We propose a simple branch-and-bound algorithm for this class of pseudo-Boolean problems and analyze its performance on a number of randomly generated instances. Acknowledgements: We are deeply thankful to Alex Kogan, for helpful discussions on a preliminary version of this text. The first author would like to acknowledge the generous support of a DIMACS Graduate Student Award.

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