Pruning moves

The concept of dominance among nodes of a branch-and-bound tree, although known for a long time, is typically not exploited by general-purpose mixed-integer linear programming (MILP) codes. The starting point of our work was the general-purpose dominance procedure proposed in the 1980s by Fischetti and Toth, where the dominance test at a given node of the branch-and-bound tree consists of the (possibly heuristic) solution of a restricted MILP only involving the fixed variables. Both theoretical and practical issues concerning this procedure are analyzed, and important improvements are proposed. In particular, we use the dominance test not only to fathom the current node of the tree, but also to derive variable configurations called ‘nogoods’ and, more generally, ‘improving moves.’ These latter configurations, which we rename ‘pruning moves’ so as to stress their use in a node-fathoming context, are used during the enumeration to fathom large sets of dominated solutions in a computationally effective way. Computational results on a testbed of MILP instances whose structure is amenable to dominance are reported, showing that the proposed method can lead to a considerable speedup when embedded in a commercial MILP solver.