Finding k edge-disjoint spanning trees of minimum total weight in a network: An application of matroid theory

Abstract

The by now classical Held and Karp procedure for the travelling salesman problem (TSP) and the “3”/2-heuristic of Christofides for the Euclidian TSP are both based on the existence of good algorithms for the minimum spanning tree problem.

The problem of finding k edge-disjoint Hamiltonian circuits of minimum total weight in a network, k≥2, (by J. Krarup called the peripatetic salesman problem (PSP)), is related to problems of both practical and theoretical importance (reallocation of governmental institutions in Sweden, vulnerability in networks). Trying to generalize the Held and Karp procedure and the “3”/2-heuristic to solve the PSP, the problem of finding k edge-disjoint spanning trees of minimum total weight in a network (k-MSTP) arises. This problem can be formulated as finding a minimum weight base in a matroid and hence the greedy algorithm can be applied if appropriate independence testing routines are available.

In this paper, we first introduce the necessary concepts and notation from matroid theory including the sum of matroids, and giving a non-standard proof we establish that the sum of k matroids is a matroid.

By means of the sum of matroids, the k-MSTP is formulated as a matroid problem, and two independence testing routines (both variants of the matroid partition algorithm of J. Edmonds) for the matroid in question are described. These are compared w.r.t. computational complexity and computational behaviour, in the latter case with special emphasis on k-MSTP for large sparse graphs.

Finally, the difficulties arising when applying the above sketched exact and heuristic methods to the PSP are discussed.