Optimal Decision Trees for the Algorithm Selection Problem: Integer Programming Based Approaches

Even though it is well known that for most relevant computational problems different algorithms may perform better on different classes of problem instances, most computational experiments still focus on determining a single best algorithm configuration based on aggregate results such as the average. In this paper, we propose Integer Programming based approaches to build decision trees for the Algorithm Selection Problem. These techniques allow to automatically: (i) find the most important problem features to determine problem classes; (ii) group the problems into classes and (iii) select the best algorithm configuration for each class. To evaluate this new approach, extensive computational experiments were executed using the linear programming algorithms implemented in the COIN-OR Branch & Cut solver in a comprehensive set of instances, including all MIPLIB benchmark instances. The results exceeded our initial expectations. While the single best parameter setting discovered decreased the total running time by 22%, our approach decreased the total running time by 40% in average in 10-fold cross validation experiments. These results indicate that our method generalizes quite well and does not overfit.

Generic CP-Supported CMSA for Binary Integer Linear Programs

Construct, Merge, Solve and Adapt (CMSA) is a general hybrid metaheuristic for solving combinatorial optimization problems. At each iteration, CMSA (1) constructs feasible solutions to the tackled problem instance in a probabilistic way and (2) solves a reduced problem instance (if possible) to optimality. The construction of feasible solutions is hereby problem-specific, usually involving a fast greedy heuristic. The goal of this paper is to design a problem-agnostic CMSA variant whose exclusive input is an integer linear program (ILP). In order to reduce the complexity of this task, the current study is restricted to binary ILPs. In addition to a basic problem-agnostic CMSA variant, we also present an extended version that makes use of a constraint propagation engine for constructing solutions. The results show that our technique is able to match the upper bounds of the standalone application of CPLEX in the context of rather easy-to-solve instances, while it generally outperforms the standalone application of CPLEX in the context of hard instances. Moreover, the results indicate that the support of the constraint propagation engine is useful in the context of problems for which finding feasible solutions is rather difficult.