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Matching: Arc Routing and the Solution Connection

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Arc Routing

Abstract

Given an undirected graph G = (V, E), a matching M ⊂ E is a subset of edges no two of which are incident with a common vertex. For any M ⊂ E, we define V(M) as the set of vertices incident to some edge in M. A matching M in G is called a maximum cardinality matching in G if ∣M∣ ≥ ∣M′∣ for all matchings M′in G. A perfect matching is a matching M with V(M) =V. Note that for the existence of perfect matchings ∣V∣ has to be an even number.

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References

  1. Altinkemer, K., and B. Gavish [1991]: Parallel savings based heuristics for the delivery problem; Operations Research 39, 456–469.

    Article  MathSciNet  MATH  Google Scholar 

  2. Applegate, D., and W. Cook [1993]: Solving large-scale matching problems; in: Network Flows and Matching: First DIMACS Implementation Challenge (D.S. Johnson and C.C. McGeoch, editors), American Mathematical Society, 557–576.

    Google Scholar 

  3. Bachem, A., and M. Malich [1993]: The simulated trading heuristic for solving vehicle routing problems; Operations Research 93, 16–19.

    Google Scholar 

  4. Balinski, M.L. [1972]: Establishing the matching polytope; J. Comb. Theory, Ser. B 13, 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  5. Ball, M.O., L.D. Bodin, and R. Dial [1983]: A matching based heuristic for scheduling mass transit crews and vehicles; Transportation Science 17, 4–31.

    Article  Google Scholar 

  6. Ball, M.O., and U. Derigs [1983]: An analysis of alternate strategies for implementing matching algorithms; Networks 13, 517–549.

    Article  MathSciNet  MATH  Google Scholar 

  7. Ball, M.O., U. Derigs, C. Hilbrand, and A. Metz [1990]: Matching problems with generalized upper bound side constraints; Networks 20, 703–721.

    Article  MathSciNet  MATH  Google Scholar 

  8. Bendisch, J., U. Derigs, and A. Metz [1994]: An efficient matching algorithm applied in statistical physis; Discrete Applied Mathematics 52, 139–153.

    Article  MathSciNet  MATH  Google Scholar 

  9. Berge, C. [1957]: Two theorems in graph theory; Proc. Natl. Acad. Sci. USA, 43, 842–844.

    Article  MathSciNet  MATH  Google Scholar 

  10. Bodin, L. and B. Golden [1981]: Classification in Vehicle Routing and Scheduling; Networks 11, 97–108.

    Article  Google Scholar 

  11. Burkard, R.E., and U. Derigs [1980]: Assignment and matching problems: Solution methods with FORTRAN-programs; Springer Lecture Notes in Mathematical Systems No. 184.

    Google Scholar 

  12. Carraresi, P., and G. Gallo [1984]: Network models for vehicle and crew scheduling; EJOR 16, 139–151.

    Article  MathSciNet  MATH  Google Scholar 

  13. Christofides, N. [1976]: Worst-case analysis of a new heuristic for the traveling salesman problem; Management Science Research Report No. 388, Carnegie-Mellon University.

    Google Scholar 

  14. Cook, W., and A. Rohe [1998]: Computing minimum weight perfect matchings; to appear in: ORSA J. on Computing.

    Google Scholar 

  15. Cornuejols, G., and G.L. Nemhauser [1978]: Tight bounds for Christofides traveling salesman heuristic; Math. Programming 14, 116–121.

    Article  MathSciNet  MATH  Google Scholar 

  16. Cunningham, W.H., and A.B. Marsh [1976]: A primal algorithm for optimal matching; Math. Programming Study 8, 50–72.

    Article  MathSciNet  Google Scholar 

  17. Derigs, U. [1981a]: A shortest augmenting path method for solving minimal perfect matching problems; Networks 11, 379–390.

    Article  MathSciNet  MATH  Google Scholar 

  18. Derigs, U. [1981b]: On some experiments with a composite heuristic for solving traveling salesman problems; Methods of Operations Research 40, 287–290.

    MATH  Google Scholar 

  19. Derigs, U. [1981c]: Another composite heuristic for solving the traveling salesman problem; University of Maryland, Working Paper.

    Google Scholar 

  20. Derigs, U. [1982]: Shortest augmenting paths and sensitivity analysis for optimal matchings; Report 82222-OR, Institut für Ökonometrie und Operations Research, Universität Bonn.

    Google Scholar 

  21. Derigs, U. [1985]: The shortest augmenting path method for solving assignment problems — Motivation and computational experience; Annals of Operations Research 4, 57–102.

    Article  MathSciNet  Google Scholar 

  22. Derigs, U. [1986]: Solving large-scale matching problems efficiently — A new primal matching approach; Networks 16, 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  23. Derigs, U. [1988a]: Programming in networks and graphs — On the combinatorial background and near-equivalence of network flow and matching algorithms; Lecture Notes in Economics and Mathematical Systems 300, Springer-Verlag.

    Google Scholar 

  24. Derigs, U. [1988b]: Solving non-bipartite matching problems via shortest path techniques; Annals of Operations Research 13, 225–261.

    Article  MathSciNet  Google Scholar 

  25. Derigs, U. and A. Metz [1986a]: On the use of optimal fractional matchings for solving the (integer) matching problem; Computing 36, 263–270.

    Article  MathSciNet  MATH  Google Scholar 

  26. Derigs, U. and A. Metz [1986b]: An in-core/out-of-core method for solving large scale assignment problems; Zeitschrift für Operations Research 30, A181–A195.

    MathSciNet  Google Scholar 

  27. Derigs, U., and A. Metz [1991]: Solving (large scale) matching problems combinatorially; Math. Programming 50, 113–121.

    Article  MathSciNet  MATH  Google Scholar 

  28. Derigs, U., and A. Metz [1992a]: Über die Matching Relaxation für das Set Partitioning Problem; in: Operations Research Proceedings 1991, 398–406.

    Google Scholar 

  29. Derigs, U., and A. Metz [1992b]: Matching problems with knapsack side constraints — A computational study; in: Modern Methods of Optimization (ed. by W. Krabs and J. Zowe), Springer Lecture Notes in Economics and Mathematical Systems 378, 48–89.

    Google Scholar 

  30. Derigs, U., and A. Metz [1992c]: A matching based approach for solving a delivery/pick-up vehicle routing problem with time constraints; OR Spektrum 14, 91–106.

    Article  MathSciNet  MATH  Google Scholar 

  31. Desrochers, M., and T.W. Verhoog [1991]: A new heuristic for the fleet size and mix vehicle routing problem; Computers and Operations Research 18, 263–274.

    Article  MATH  Google Scholar 

  32. Dijkstra, E.W. [1959]: A note on two problems in connection with graphs; Numerische Mathematik 1, 269–271.

    Article  MathSciNet  MATH  Google Scholar 

  33. Dror, M., and L. Levy [1986]: A vehicle routing improvement algorithm comparison of a ”greedy” and matching implementation for inventory routing; Computers and Operations Research 13, 33–45.

    Article  MATH  Google Scholar 

  34. Edmonds, J. [1965a]: Paths, trees and flowers; Can J. Math. 17, 449–467.

    Article  MathSciNet  MATH  Google Scholar 

  35. Edmonds, J. [1965b]: Maximum matching and a polyhedron with 0,1 vertices; J. Res. Natl. Bur. Standards 69B, 125–130.

    Article  MathSciNet  Google Scholar 

  36. Edmonds, J., and E.L. Johnson [1973]: Matching, Euler tours and the Chinese postman; Math. Programming 5, 88–124.

    Article  MathSciNet  MATH  Google Scholar 

  37. Edmonds, J. and W. Pulleyblank [1974]: Facets of 1-matching polyhedra; in: Hypergraph Seminar, Lecture Notes in Mathematics, 411, 214–242.

    Article  MathSciNet  Google Scholar 

  38. Euler, L. [1736]: Solutio problematis ad geometriam situs pertinentis; Comment. Acad. Sci. Imp. Petropolitanae 8, 128–140.

    Google Scholar 

  39. Fujii, M., T. Kasami, and N. Ninomiya [1969]: Optimal sequencing of two equivalent processors; SIAM J. Appl. Math 17, 784–789 [Erratum in: SIAM J. Appl. Math. 20 (1971) 141].

    Article  MathSciNet  MATH  Google Scholar 

  40. Gabow, H.N. [1976]: An efficient implementation of Edmond’ s algorithm for maximum matching on graphs; J. of the ACM 23, 221–234.

    Article  MathSciNet  MATH  Google Scholar 

  41. Gabow, H.N. [1990]: Data structes for weighted matching and nearest common ancestors with linking; Proc. of the First Annual ACM-SIAM Symp. on Discrete Algorithms, ACM, New York, 434–443.

    Google Scholar 

  42. Gabow, H.N., and R.E. Tarjan [1983]: A linear-time algorithm for a special case of disjoint set union; Proc. 15th Annual ACM Symp. on Theory of Computing, York, N.Y., 246–251.

    Google Scholar 

  43. Gabow, H.N., and R.E. Tarjan [1991]: Faster scaling algorithms for general graph matching problems; J. of the ACM 38, 815–853.

    Article  MathSciNet  MATH  Google Scholar 

  44. Gerards, A.M.H. [1995]: Matching; in: Handbooks in OR & MS 7 (M.O. Ball, T.L. Magnanti, C.L. Monma, and G.L. Nemhauser, editors), North Holland, Amsterdam.

    Google Scholar 

  45. Grötschel, M. [1977]: Polyedrische Charakterisierung Kombinatorischer Optimierungsprobleme; Mathematical Systems in Economics 36; Verlag Anton Hain, Meisenheim am Glan.

    Google Scholar 

  46. Grötschel, M., and O. Holland [1985]: Solving matching problems with linear programming; Math. Programming 33, 243–259.

    Article  MathSciNet  MATH  Google Scholar 

  47. Hasselström, D. [1976]: Connecting bus-routes at a point of intersection; AB VOLVO Working Paper.

    Google Scholar 

  48. Held, M., and R.M. Karp [1971]: The traveling salesman problem and minimum spanning trees: Part II; Math. Programming 1, 6–25.

    Article  MathSciNet  MATH  Google Scholar 

  49. Iri, M., K. Murota, and S. Matsui [1983]: An approximate solution for the problem of optimizing the plotter pen movement; in: R.F. Drenick and F. Kozin (eds), ”System Modeling and Optimization”, Proc. 10th IFIP Conf., New York, 1981, Lecture Notes in Control and Information Sciences, Vol. 38, Springer-Verlag, Berlin, 572–580.

    Google Scholar 

  50. Johnson, D.S., and C.C. McGeoch [1993]: Network Flows and Matching — First DIM ACS Implementation Challenge; American Mathematical Society.

    Google Scholar 

  51. Kruskal Jr., J.B. [1956]: On the shortest spanning subtree of a graph and the traveling salesman problem; Proc. Amer. Math. Soc. 7, 48–50.

    Article  MathSciNet  MATH  Google Scholar 

  52. Kuhn, H.W. [1955]: The Hungarian method for the assignment problem; Nav. Res. Log. Quart. 2, 83–97.

    Article  Google Scholar 

  53. Kwan, Mei Ko [1962]: Graphic programming using odd and even points; Chinese Math. 1, 273–277.

    MathSciNet  MATH  Google Scholar 

  54. Lawler, E.L. [1976]: Combinatorial Optimization: Networks and Matroids; Holt, Rinehart, and Winston, New York, N.Y.

    MATH  Google Scholar 

  55. Lin, S., and B.W. Kernighan [1973]: An effective heuristic algorithm for the traveling salesman problem; Operations Research 21, 498–516.

    Article  MathSciNet  MATH  Google Scholar 

  56. Lovasz, L., and M.D. Plummer [1986]: Matching Theory; Annals of Discrete Mathematics 29, North Holland.

    Google Scholar 

  57. Machol, R.E. [1961]: An application of the assignment problem; Operations Research 9, 585–586.

    Article  Google Scholar 

  58. Metz, A. [1987]: Postoptimale Analyse und neue primale Matching Algorithmen; Diploma Thesis, University of Cologne.

    Google Scholar 

  59. Miller, D.L. [1995]: A matching based exact algorithm for capacitated vehicle routing problems; ORSA J. on Computing 7, 1–9.

    Article  MATH  Google Scholar 

  60. Nemhauser, G., and G. Weber [1979]: Optimal set partitioning, matchings and Lagrangean duality; Naval Res. Log. Quart. 26, 553–563.

    MathSciNet  MATH  Google Scholar 

  61. Olafson, S. [1990]: Weighted matching in chess tournaments; Journal of the Operational Research Society 41, 17–24.

    Google Scholar 

  62. Pandit, R. and B. Muraldharan [1995]: A capacitated general routing problem on mixed networks; Computers and Operations Research 26, 465–478.

    Article  Google Scholar 

  63. Papadimitriou, C.H., and K. Steiglitz [1976]: Combinatorial Optimization: Algorithms and Complexity; Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  64. Reingold, E.M., and R.E. Tarjan [1981]: On a greedy heuristic for complete matching; SIAM J. Comput. 10, 676–681.

    Article  MathSciNet  MATH  Google Scholar 

  65. Riskin, E., R. Ladner, R. Wand, and L. Atlas [1994]: Index assignment for progressive transmission of full-search vector quantization; IEEE Transactions on Image Processing 3, 307–312.

    Article  Google Scholar 

  66. Schrijver, A. [1983]: Short proofs on the matching polyhedron; J. Comb. Theory, Ser. B 34, 104–108.

    Article  MathSciNet  MATH  Google Scholar 

  67. Tarjan, R.E. [1983]: Data structures and network algorithms; SIAM. Philadelphia.

    Book  Google Scholar 

  68. Vaidya, P.M. [1989]: Geometry helps in matching; SIAM J. Comput. 18, 1201–1225.

    Article  MathSciNet  MATH  Google Scholar 

  69. Weber, G. [1981]: Sensitivity analysis of optimal matchings; Networks 11, 41–56.

    Article  MathSciNet  MATH  Google Scholar 

  70. Win, Z. [1988]: Contributions to routing problems; Unpublished Ph.D. dissertation Universität Ausburg.

    Google Scholar 

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Authors and Affiliations

  1. University of Cologne, Germany

    Ulrich Derigs

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  1. Ulrich Derigs
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Editors and Affiliations

  1. Department of Management Information Systems, University of Arizona, Tucson, Arizona, 85721, USA

    Moshe Dror

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Derigs, U. (2000). Matching: Arc Routing and the Solution Connection. In: Dror, M. (eds) Arc Routing. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4495-1_3

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  • DOI: https://doi.org/10.1007/978-1-4615-4495-1_3

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