Solving a "Hard" problem to approximate an "Easy" one: heuristics for maximum matchings and maximum traveling salesman problems

Authors:
Sándor P. Fekete;Henk Meijer;André Rohe;Walter Tietze
Affiliations:
Department of Mathematical Optimization, TU Braunschweig, 38106 Braunschweig, GERMANY;Department of Computing and Information Science, Queen's University, Kingston, Ontario K7L 3N6, CANADA;Forschungsinstitut für Diskrete Mathematik, Universität Bonn, 53113 Bonn, GERMANY;Department of Mathematics, TU Berlin, 10623 Berlin, GERMANY
Venue:
Journal of Experimental Algorithmics (JEA)
Year:
2002

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Abstract

We consider geometric instances of the Maximum Weighted Matching Problem (MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000 vertices. Making use of a geometric duality relationship between MWMP, MTSP, and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields in near-linear time solutions as well as upper bounds. Using various computational tools, we get solutions within considerably less than 1% of the optimum.An interesting feature of our approach is that, even though an FWP is hard to compute in theory and Edmonds' algorithm for maximum weighted matching yields a polynomial solution for the MWMP, the practical behavior is just the opposite, and we can solve the FWP with high accuracy in order to find a good heuristic solution for the MWMP.