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Solving matching problems with linear programming. (English) Zbl 0579.90069

Summary: We describe an implementation of a cutting plane algorithm for the perfect matching problem which is based on the simplex method. The algorithm has the following features: 1. It works on very sparse subgraphs of K n which are determined heuristically, global optimality is checked using the reduced cost criterion. 2. Cutting plane recognition is usually accomplished by heuristics. Only if these fail, the Padberg-Rao procedure is invoked to guarantee finite convergence.

Our computational study shows that - on the average - very few variables and very few cutting planes suffice to find a globally optimal solution. We could solve this way matching problems on complete graphs with up to 1000 nodes. Moreover, it turned out that our cutting plane algorithm is competitive with the fast combinatorial matching algorithms known to date.

MSC:
90C10Integer programming
90C05Linear programming
68Q25Analysis of algorithms and problem complexity
65K05Mathematical programming (numerical methods)
References:
[1]M. Ball and U. Derigs, ”An analysis of alternate strategies for implementing matching algorithms”,Networks 13 (1983) 517–549. · Zbl 0519.68055 · doi:10.1002/net.3230130406
[2]R.E. Burkard and U. Derigs,Assignment and Matching Problems: Solution Methods with Fortran-Programs (Springer Lecture Notes in Economics and Mathematical Systems, No. 184, Berlin, 1980).
[3]H. Crowder and M.W. Padberg, ”Solving large-scale symmetric travelling salesman problems”,Management Science 26 (1980) 495–509. · Zbl 0444.90068 · doi:10.1287/mnsc.26.5.495
[4]W. Cunningham and A. Marsh, ”A primal algorithm for optimum matching”,Mathematical Programming Study 8 (1978) 50–72.
[5]U. Derigs, ”Solving matching problems via shortest path techniques”, Report No. 83263-OR, Institut für Ökonometrie und Operations Research, Universität Bonn (Bonn, 1983).
[6]U. Derigs, ”Solving large scale matching problems efficiently–A new primal matching approach”, Report No. 84346-OR, Institut für Ökonometrie und Operations Research, Universität Bonn (Bonn, 1984).
[7]E.A. Dinic, ”Algorithm for solution of a problem of maximum flow in a network with power estimation”,Soviet Mathematics Doklady 11 (1970) 1277–1280.
[8]J. Edmonds, ”Paths, trees and flowers”,Canadian Journal of Mathematics 17 (1965) 449–467. · Zbl 0132.20903 · doi:10.4153/CJM-1965-045-4
[9]J. Edmonds, ”Maximum matching and a polyhedron with 0, 1 vertices”,Journal of Research National Bureau of Standards 69B (1965) 125–130.
[10]L.R. Ford and D.R. Fulkerson, ”A simple algorithm for finding maximal flows and an application to the Hitchcock problem”,Canadian Journal of Mathematics 9 (1957) 210–218. · Zbl 0088.12907 · doi:10.4153/CJM-1957-024-0
[11]F. Glover, D. Klingman, J. Mote and D. Whitman, ”A primal simplex variant for the maximum flow problem”, Center for Cybernetic Studies, CCS 362 (Austin, TX, 1979).
[12]M. Grötschel, M. Jünger and G. Reinelt, ”A cutting plane algorithm for the linear ordering problem”,Operations Research 32 (1984) 1195–1220. · Zbl 0554.90077 · doi:10.1287/opre.32.6.1195
[13]M. Grötschel, L. Lovász and A. Schrijver, ”The ellipsoid method and its consequences in combinatorial optimization”,Combinatorica 1 (1981) 169–197. · Zbl 0492.90056 · doi:10.1007/BF02579273
[14]M. Grötschel and W.R. Pulleyblank, ”Weakly bipartite graphs and the max-cut problem”,Operations Research Letters 1 (1981) 23–27. · Zbl 0494.90078 · doi:10.1016/0167-6377(81)90020-1
[15]A.V. Karzanov, ”Determining the maximal flow in a network by the method of preflows”,Soviet Mathematics Doklady 15 (1972) 434–437.
[16]E.L. Lawler,Combinatorial optimization: Networks and matroids (Holt, Rinehart and Winston, New York, 1976).
[17]V.M. Malhorta, M.P. Kumar and S.N. Maheshwari, ”An O(|V|3) algorithm for finding maximum flows in networks”,Information Processing Letters 7 (1978) 277–278. · Zbl 0391.90041 · doi:10.1016/0020-0190(78)90016-9
[18]M.W. Padberg and M.R. Rao, ”Odd minimum cut-sets andb-matchings”,Mathematics of Operations Research 7 (1982) 67–80. · Zbl 0499.90056 · doi:10.1287/moor.7.1.67
[19]M.W. Padberg and M.R. Rao, ”The Russian method for linear inequalities III: Bounded integer Programming”, Preprint, GBA New York University, New York, May 1981.