, Volume 4, Issue 1, pp 1–53 | Cite as

Combinatorial optimization and small polytopes

  • T. Christof
  • G. Reinelt


Travel Salesman Problem Travel Salesman Problem Combinatorial Optimization Problem Fractional Solution Incidence Vector 
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  1. Alevras, D., G. Cramer and M. Padberg (1994). Double Description Algorithm for Calculating a Basis of the Lineality Space and the Extreme Rays of a Polyhedral Cone. Software available via ftp at, directory pub/mathprog/polyth/dda.Google Scholar
  2. Alevras, D. (1995). personal communication.Google Scholar
  3. Applegate, D., R.E. Bixby, V. Chvátal and W. Cook (1994). Finding cuts in the TSP (A preliminary report). Research Report, Rice University.Google Scholar
  4. Avis, D. (1993). AC Implementation of the Reverse Search Vertex Enumeration Algorithm. Software available via ftp at, directory pub/C.Google Scholar
  5. Avis, D. and K. Fukuda (1992). A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra.Discrete and Computational Geometry 8, 295–313.CrossRefGoogle Scholar
  6. Balas, E., S. Ceria and C. Cornuéjols (1993). A Lift-and-Project Cutting Plane Algorithm for Mixed 0–1 Programs.Mathematical Programming 58, 295–324.CrossRefGoogle Scholar
  7. Borchers, B. (1992). Improved Branch and Bound Algorithms for Integer Programming. Troy, NY.Google Scholar
  8. Boyd, S.C. and W.H. Cunningham (1991). Small Traveling Salesman Polytopes.Mathematics of Operations Research 16, 259–271.Google Scholar
  9. Bruger, E. (1956). Über homogene lineare Ungleichungssysteme.Zeitschrift für Angewandte Mathematik und Mechanik 36, 135–139.Google Scholar
  10. Chernikov, S.N. (1971).Lineare Ungleichungen. Berlin: Deutscher Verlag der Wissenschaften.Google Scholar
  11. Chernikova, N.V. (1965). Algorithm for Finding a General Formula for the Nonnegative Solutions of a System of Linear Inequalities.U.S.S.R. Computational Mathematics and Mathematical Physics 5, 228–233.CrossRefGoogle Scholar
  12. Christof, T. (1991).Ein Verfahren zur Transformation zwischen Polyederdarstellungen. Diploma Thesis, Universität Augsburg.Google Scholar
  13. Christof, T., M. Jünger and G. Reinelt (1991). A Complete Description of the Traveling Salesman Polytope on 8 Nodes.Operations Research Letters 10, 497–500.CrossRefGoogle Scholar
  14. Christof, T. and G. Reinelt (1995). Parallel Cutting Plane Generation for the TSP, in P. Fritzson and L. Finmo, eds.Parallel Programming and Applications. IOS Press.Google Scholar
  15. Dantzig, G.B. and B.C. Eaves (1973). Fourier-Motzkin Elimination and its Dual.J. Combinatorial Theory Ser. A 14, 228–297.CrossRefGoogle Scholar
  16. Dantzig, G.B., R. Fulkerson and S.M. Johnson (1954). Solution of a Large-Scale Traveling Salesman Problem.Operations Research 2, 393–410.Google Scholar
  17. Duffin, R.J. (1974). On Fourier's Analysis of Linear Inequality Systems. Mathematical Programming Study 1.Google Scholar
  18. Edmonds, J. (1965). Maximum Matching and a Polyhedron with (0,1) Vertices.Journal Res. Nat. Bur. Stand. 69B, 125–130.Google Scholar
  19. Edelsbrunner, H. (1987).Algorithms in Combinatorial Geometry. Berlin: Springer.Google Scholar
  20. Euler, R. and H. Le Verge (1992). A Complete and Irredundant Linear Description of the Asymmetric Traveling Salesman Polytope on 6 Nodes. Research Report, University of Brest.Google Scholar
  21. Fishburn, P.C. (1990). Binary Probabilities Induced by Rankings. SIAM Journal of Discrete Mathematics3, 478–488.CrossRefGoogle Scholar
  22. Fishburn, P.C. (1991). Induced Binary Probabilities and the Linear Ordering Polytope: A Status Report. Report AT&T Bell Laboratories.Google Scholar
  23. Fukuda, K. (1993). cdd.c.:C Implementation of the Double Desription Method for Computing All Vertices and Extremal Rays of a Convex Polyhedron Given by a System of Linear Inequalities. Technical Report, DMA-EPFL Lausanne, software available via ftp at, directory pub/fukuda/cdd.Google Scholar
  24. Fukuda, K. and A. Prodon (1995). Double Description Method Revisited. Report, ETHZ Zürich.Google Scholar
  25. Galperin, A.M. (1976). The General Solution of a Finite Set of Linear Inequalities.Math. Oper. Res,1, 185–196.Google Scholar
  26. Gomory, R.E. (1963). An Algorithm for Integer Solutions to Linear Programs, in R.L. Graves and P. Wolfe, eds.Recent Advances in Mathematical Programming. New York: McGraw Hill.Google Scholar
  27. Grötschel, M. and O. Holland (1985). Solving Matching Problems with Linear Programming.Mathematical Programming 33, 243–259.CrossRefGoogle Scholar
  28. Grötschel, M., M. Jünger and G. Reinelt (1984). A Cutting Plane Algorithm for the Linear Ordering Problem.Operations Research 32, 1195–1220.Google Scholar
  29. Grötschel, M., M. Jünger and G. Reinelt (1985). Facets of the Linear Ordering Polytope.Mathematical Programming 33, 43–60.CrossRefGoogle Scholar
  30. Grötschel, M., L. Lovász and A. Schrijver (1988).Geometric Algorithms and Combinatorial Optimization. Berlin: Springer.Google Scholar
  31. Grötschel, M. and M.W. Padberg (1985). Polyhedral theory, in E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, eds.The Traveling Salesman Problem. Chichester: Wiley & Sons.Google Scholar
  32. Jünger, M., G. Reinelt and G. Rinaldi (1995). The Traveling Salesman Problem, in M. Ball, T. Magnanti, C.L. Monma and G.L. Nemhauser, eds.Handbooks in Operations Research and Management Sciences: Networks. North-Holland.Google Scholar
  33. Jünger, M., G. Reinelt and S. Thienel (1994). Optimal and Provably Good Solutions for the Symmetric Traveling Salesman Problem.Zeitschrift für Operations Research 40, 183–217.Google Scholar
  34. Jünger, M., G. Reinelt and S. Thienel (1995). Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization, in W. Cook, L. Lovász and P. Seymour, eds.DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 20: Combinatorial Optimization. American Mathematical Society.Google Scholar
  35. Jünger, M. and P. Störmer (1995). Solving Large-Scale Traveling Salesman Problems with parallel Branch-and-Cut. Technical Report No. 95.191, Universität zu Köln.Google Scholar
  36. Kirkpatrick, C.D., Gelatt Jr. and M.P. Vecchi (1983). Optimization by Simulated Annealing.Science 222, 671–680.CrossRefGoogle Scholar
  37. Leung, J. and J. Lee (1994). “More Facets from Fences for Linear Ordering and Acyclic Subgraph Polytopes”. Discrete Appl. Math.50, No. 2, 185–200.CrossRefGoogle Scholar
  38. Mathies, T.H. and D.S. Rubin (1980). A Survey and Comparison of Methods for Finding all Vertices of Convex Polyhedral Sets.Mathematics of Operations Research 5, 167–184.Google Scholar
  39. Mitchell, J.E. (1994). Interior Point Algorithms for Integer Programming. Rensselaer Polytechnic Institute, TR 215.Google Scholar
  40. Motzkin, T.S., H. Raiffa, G.L. Thompson and R.M. Thrall (1953). The Double Description Method, in H.W. Kuhn and A.W. Tucker, eds.Contributions to the Theory of Games (Vol. 2). Princeton University Press, Princeton, N. J.Google Scholar
  41. Müller, R. (1992). On the Transitive Acyclic Subdigraph Polytope. Report No. 337, TU Berlin.Google Scholar
  42. Naddef, D. and G. Rinaldi (1992). The Crown Inequalities for the Symmetric Traveling Salesman Polytope.Mathematics of Operations Research 17, 308–326.Google Scholar
  43. Naddef D. and G. Rinaldi (1993). The Graphical Relaxation: A New Framework for the Symmetric Traveling Salesman Polytope.Mathematical Programming 58, 53–87.CrossRefGoogle Scholar
  44. Padberg, M. (1995).Linear Optimization and Extensions, Berlin: Springer.Google Scholar
  45. Padberg, M.W. and G. Rinaldi (1987). Optimization of a 532 City Symmetric Traveling Salesman Problem by Branch and Cut.Operations Research Letters 6, 1–7.CrossRefGoogle Scholar
  46. Padberg, M.W. and G. Rinaldi (1990). Facet Identification for the Symmetric Traveling Salesman Polytope.Mathematical Programming 47, 219–257.CrossRefGoogle Scholar
  47. Padberg, M.W. and G. Rinaldi (1991). A Branch and Cut Algorithm for the Resolution of Large-scale Symmetric Traveling Salesman Problems.SIAM Review 33, 60–100.CrossRefGoogle Scholar
  48. Pulleyblank, W.R. (1989). Polyhedral Combinatorics, in G.L. Nemhauser, A.H.G. Rinnoy Kan and M.J. Todd, eds.Handbooks in Operations Research and Management Sciences: Optimization. North-Holland.Google Scholar
  49. Queyranne, M. and Y. Wang (1993). Hamiltonian Path and Symmetric Traveling Salesman Polytopes.Mathematical Programming 58, 89–110.CrossRefGoogle Scholar
  50. Reinelt, G. (1985).The Linear Ordering Problem: Algorithms and Applications. Research and exposition in mathematics,8, Berlin: Heldermann.Google Scholar
  51. Reinelt, G. (1991). TSPLIB—A Traveling Salesman Problem Library.ORSA Journal on Computing 3, 376–384. (WWW-access: http:// Scholar
  52. Reinelt, G. (1993). A Note on Small Linear Ordering Polytopes.Discrete & Computational Geometry 10, 67–78.CrossRefGoogle Scholar
  53. Rubin, D. (1975). Vertex Generation and Cardinality Constrained Linear Programs.Operations Research 23, 555–565.CrossRefGoogle Scholar
  54. Schrijver, A. (1986).Theory of Linear and Integer Programming. Chichester: John Wiley & Sons.Google Scholar
  55. Suck, R. (1991). Geometric and Combinatorial Properties of the Polytope of Binary Choice Probabilities. Report Universität Osnabrück.Google Scholar
  56. Thienel, S. (1995).ABACUS—A Branch and Cut System. PhD Thesis, Universität zu Köln.Google Scholar
  57. Wilde, D. (1993). A Library for Doing Polyhedral Operations. Report 785, IRISA, Rennes, France, Software available via ftp at, directory: local/API.Google Scholar
  58. Verge Le, H. (1992). A Note on Chernikova's Algorithm. Report 785, IRISA, Rennes, France.Google Scholar
  59. Zigler, G. (1995). Lectures on Polytopes, Berlin: Springer.Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 1996

Authors and Affiliations

  • T. Christof
    • 1
  • G. Reinelt
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany

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