On the Separability of Subproblems in Benders Decompositions
Benders decomposition is a well-known procedure for solving a combinatorial optimization problem by defining it in terms of a master problem and a subproblem. Its effectiveness relies on the possibility of synthethising Benders cuts (or nogoods) that rule out not only one, but a large class of trial values for the master problem. In turns, this depends on the possibility of separating the subproblem into several subproblems, i.e., problems exhibiting strong intra-relationships and weak inter-relationships. The notion of separation is typically given informally, or relying on syntactical aspects. This paper formally addresses the notion of separability of the subproblem by giving a semantical definition and exploring it from the computational point of view. Several examples of separable problems are provided, including some proving that a semantical notion of separability is much more helpful than a syntactic one. We show that separability can be formally characterized as equivalence of logical formulae, and prove the undecidability of the problem of checking separability.
KeywordsMaster Problem Secondary Variable Bender Decomposition Semantical Notion Alldifferent Constraint
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