Abstract
The Binate Covering Problem (BCP) is the problem of finding a minimum cost assignment to variables that is a solution of a boolean equation f = 1. It is a generalization of the set covering (or unate covering) problem, where f is positive unate, and is generally given as a table with rows corresponding to the set elements and the columns corresponding to the subsets.
Previous methods have considered the case when f is given as a product-of-sum formula or as a binary decision diagram (BDD). In this paper we present a new branchand-bound algorithm for the BCP, that assumes f is expressed as the conjunction of multiple BDD’s.
In general all 0-1 integer linear programs can be translated into a binate covering problem. However, if the characteristic function is represented as a product of sums, the number of clauses may exceed the number of linear constraints by so far as to render the method impractical. On the contrary, the representation by means of BDD’s gives one BDD per linear constraint and these BDD’s are generally well behaved. Hence the new BDD-based algorithm is suited as a general solver of 0-1 linear programs. In particular, we have applied our BCP solver to the exact minimization of boolean relations and we have been able to solve difficult binate covering problems with thousands of variables (the larger problem we have solved so far has over 4600 variables).
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Jeong, SW., Somenzi, F. (1993). A New Algorithm for 0-1 Programming Based on Binary Decision Diagrams. In: Sasao, T. (eds) Logic Synthesis and Optimization. The Kluwer International Series in Engineering and Computer Science, vol 212. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3154-8_7
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