Optimal cost partitioning of classical planning heuristics has been shown to lead to excellent heuristic values but is often prohibitively expensive to compute. We analyze the application of Lagrangian decomposition, a classical tool in mathematical programming, to cost partitioning of operator-counting heuristics. This allows us to view the computation as an iterative process that can be seeded with any cost partitioning and that improves over time. In the case of non-negative cost partitioning of abstraction heuristics the computation reduces to independent shortest path problems and does not require an LP solver.

The split closure has been proved in practice to be a very tight approximation of the integer hull formulation of a generic mixed-integer linear program. However, exact separation procedures for optimizing over the split closure have unacceptable computing times in practice; hence, many different heuristic strategies have been proposed in the last few years. In this paper we present a new overall framework for approximating the split closure that merges different ideas from the previous approaches. Computational results prove the effectiveness of the proposed procedure compared to the state of the art, showing that a good approximation of the split closure bound can be obtained with very reasonable computing times.

We study the computational complexity of finding the next most preferred solution in some common formalisms for representing constraints and preferences. The problem is computationally intractable for CSPs, but is polynomial for tree-shaped CSPs and tree-shaped fuzzy CSPs. On the other hand, it is intractable for weighted CSPs, even under restrictions on the constraint graph. For CP-nets, the problem is polynomial when the CP-net is acyclic. This remains so if we add (soft) constraints that are tree-shaped and topologically compatible with the CP-net.

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.