Network flow models for designing diameter-constrained minimum-spanning and Steiner trees

We formulate and computationally test several models for the Diameter-Constrained Minimum Spanning and Steiner Tree Problems, which seek a least-cost spanning or Steiner tree subject to a (diameter) bound imposed on the number of edges in the tree between any node pair. A traditional multicommodity flow model with a commodity for every pair of nodes was unable to solve a 20-node and 100-edge spanning tree problem after 1 week of computation. In contrast, the new models were able to optimality solve this problem in less than 1 second and larger problem instances with up to 100 nodes and 1000 edges. The largest model contains more than 250,000 integer variables and more than 125,000 constraints. The new models simultaneously find a directed tree with a central node or a central edge that serves as a source for the commodities in a multicommodity flow model with hop constraints. Our results demonstrate the power of using single-sourcing combined with other reformulation techniques: directing the model and using hop-indexed formulations. Enhancements improve the models when the diameter bound is odd (these situations are more difficult to solve). The linear programming relaxation of the best formulations discussed in this paper always give an optimal integer solution for two special, polynomially solvable cases of the problem.