Practical optimization of Steiner trees via the cavity method

Abstract

The optimization version of the cavity method for single instances, called Max-Sum, has been applied in the past to the minimum Steiner tree problem on graphs and variants. Max-Sum has been shown experimentally to give asymptotically optimal results on certain types of weighted random graphs, and to give good solutions in short computation times for some types of real networks. However, the hypotheses behind the formulation and the cavity method itself limit substantially the class of instances on which the approach gives good results (or even converges). Moreover, in the standard model formulation, the diameter of the tree solution is limited by a predefined bound, that affects both computation time and convergence properties. In this work we describe two main enhancements to the Max-Sum equations to be able to cope with optimization of real-world instances. First, we develop an alternative ‘flat’ model formulation that allows the relevant configuration space to be reduced substantially, making the approach feasible on instances with large solution diameter, in particular when the number of terminal nodes is small. Second, we propose an integration between Max-Sum and three greedy heuristics. This integration allows Max-Sum to be transformed into a highly competitive self-contained algorithm, in which a feasible solution is given at each step of the iterative procedure. Part of this development participated in the 2014 DIMACS Challenge on Steiner problems, and we report the results here. The performance on the challenge of the proposed approach was highly satisfactory: it maintained a small gap to the best bound in most cases, and obtained the best results on several instances in two different categories. We also present several improvements with respect to the version of the algorithm that participated in the competition, including new best solutions for some of the instances of the challenge.