Combinatorial Optimization II pp 88-101 | Cite as
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.
Key words
Circuits Edge-disjoint Graph Greedy Algorithm Hamiltonian Circuit Heuristic Matroid (Partition) Minimum Spanning Tree (Peripatetic) Travelling SalesmanPreview
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References
- [1]A.V. Aho, J.E. Hopcroft and J.D. Ullman, The design and analysis of computer algorithms (Addison-Wesley, Reading, MA, 1974).MATHGoogle Scholar
- [2]N. Christofides, “Worst-case analysis of a new heuristic for the travelling salesman problem”, Management science report no. 388, Carnegie-Mellon University (1976).Google Scholar
- [3]N. Christofides and C. Whitlock, “Graph connectivity and vulnerability, a survey”, Manuscript presented at the summer school on combinatorial optimization, Urbino, Italy (1978).Google Scholar
- [4]J. Clausen, “Matroids and combinatorial optimization”, Report no. 78/4, Institute of Datology, University of Copenhangen, Denmark (1978).Google Scholar
- [5]J. Clausen and T. Hoholdt, “On the sum of matroids”, Research report, Institute of Mathematics, Technical University of Denmark (1975).Google Scholar
- [6]J. Edmonds, “Minimum partition of a matroid into independent subsets”, Journal of the National Bureau of Standards 69B (1965) 67–72.MathSciNetGoogle Scholar
- [7]F. Glover, D. Klingman and J. Stuts, “Augmented threaded index method for network optimization”, Journal of Operational Research and Information Processing 12 (1974) 293–298.MATHGoogle Scholar
- [8]K.H. Hansen and J. Krarup, “Improvements of the Held-Karp algorithm for the symmetric travelling salesman problem”, Mathematical Programming 7 (1975) 87–96.CrossRefGoogle Scholar
- [9]M. Held and R.M. Karp, “The travelling salesman problem and minimum spanning trees: Part II”, Mathematical Programming 1 (1971) 6–25.MATHCrossRefMathSciNetGoogle Scholar
- [10]D. Knuth, “Matroid partitioning”, Research report no. STAN-CS-73-342, Stanford University (1973).Google Scholar
- [11]J. Krarup, “The peripatetic salesman and some related unsolved problems”, in: B. Roy, ed., Combinatorial programming: methods and applications (D. Reidel Publishing Company, Dordrecht, 1975) pp. 173–178.Google Scholar
- [12]E.L. Lawler, Combinatorial optimization: networks and matroids (Holt, Rinehart and Winston, New York, 1976).MATHGoogle Scholar
- [13]G.H.J. Meredith, “Regular n-valent n-connected non Hamiltonian non-edge-colorable graphs”, Journal of Combinatorial Theory 14(B) (1973) 55–60.MATHCrossRefMathSciNetGoogle Scholar
- [14]L. Mirsky, Transversal theory (Academic Press, London, 1971).MATHGoogle Scholar
- [15]B. Petersen, “Investigating solvability and complexity of linear active networks by means of matroids”, Research report, Institute of Mathematics, Technical University of Denmark (1977).Google Scholar
- [16]B. Petersen, “Circuits in the union of matroids: an algorithmic approach”, Research report. Institute of Mathematics, Technical University of Denmark (1978).Google Scholar
- [17]D.J.A. Welsh, Matroid Theory (Academic Press, London, 1976).MATHGoogle Scholar