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LP-Based Combinatorial Problem Solving

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Computational Mathematical Programming

Part of the book series: NATO ASI Series ((NATO ASI F,volume 15))

Abstract

A tutorial outline of the polyhedral theory that underlies linear-programming (LP)-based combinatorial problem solving is given Design aspects of a combinatorial problem solver are discussed in general terms. Three computational studies in combinatorial problem solving using the polyhedral theory developed in the past fifteen years are surveyed: one addresses the symmetric traveling salesman problem, another the optimal triangulation of input/output matrices and the third the optimization of large-scale zero-one linear programming problems.

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References

  • Balas, E. & R. Martin [1980]: “Pivot and Complement-A Heuristic for 0-1 Programming” Management Science 26 (1980) 86–96.

    Article  MATH  MathSciNet  Google Scholar 

  • Baranyi, I., T. Van Roy, & L. Wolsey, [1983a]: “Strong Formulations for Multi-Item Capacitated Lot Sizing,” CORE Report No. 8313, Universite Catholique, Louvain-la-Neuve, Belgium.

    Google Scholar 

  • Baranyi, I., T. Van Roy, & L. Wolsey, [1983b]: “Uncapacitated, Lot-Sizing: The Convex Hull of Solutions,” CORE Report No. 8314, Universite Catholique, Louvain-la-Neuve, Belgium.

    Google Scholar 

  • Burkard, R. & U. Derigs, [1980]: Assignment and Matching Problems: Solution Methods with FORTRAN-Programs, Springer Lecture Notes in Economics and Mathematical Systems, No. 184, Berlin.

    Google Scholar 

  • Cho, Y. D. Chinhyung, E.L. Johnson, M. Padberg M.R. Rao [1982]: “On the Uncapacitated Plant Location Problem I: Valid Inequalities and Facets,” Mathematics of Operations Research 8 (1983) 579–589.

    Article  MATH  MathSciNet  Google Scholar 

  • Cho, Y. D. Chinhyung, M. Padberg & M.R. Rao, [1983]: “On the Uncapacitated Plant Location Problem II: Facets and Lifing Theorems,” Mathematics of Operations Research 8 (1983) 590–612.

    Article  MATH  MathSciNet  Google Scholar 

  • Chvatal, V. [1973]: “Edmonds Polytopes and Weakly Hamiltonian Graphs,” Mathematical Programming, 5(1973) 29–40.

    Article  MATH  MathSciNet  Google Scholar 

  • Crowder, H. & M. Padberg [1980]: “Solving Large-Scale Symmetric Traveling Salesman Problems to Optimality,” Management Science, 26 (1980) 495–509.

    Article  MATH  MathSciNet  Google Scholar 

  • Crowder, H., E. Johnson, & M. Padberg [1983]: “Solving Large-Scale Linear Zero-One Programming Problems,” Operations Research, 31 (1983) 803–834.

    Article  MATH  Google Scholar 

  • Dantzig, G., D.R. Fulkerson & S. Johnson [1954]: “Solution of a Large-Scale Traveling Salesman Problem” Operations Research 2(1954) 293-410.

    Google Scholar 

  • Edmonds, J. [1965]: “Maximum Matching and a Polyhedron with 0,1 Vertices,” Journal of Research, National Bureau of Standards, 69B (1965) 125-130.

    Google Scholar 

  • Fleischmann, B. [1981]: “The Traveling Salesman Problem on a Road Network,” Working paper, Fachbereich Wirtschaftswissenschaften, Universitat Hamburg, revised July 1982.

    Google Scholar 

  • Fleischmann, B., [1982]: “Linear-Programming Approaches to Traveling Salesman and Vehicle Scheduling Problems,” paper presented at the XI. International Symposium on Mathematical Programming, Bonn, FRG.

    Google Scholar 

  • Gomory, R. & T.C. Hu [1961]: “Multi-terminal Network Flows,” J. SIAM 9 (1961) 551–570.

    MATH  MathSciNet  Google Scholar 

  • Grotschel, M. & O. Holland, 1984]: “Solving Matching Problems with Linear Programming,” Preprint No. 37, Mathematisches Institut, Universitat Augsburg, FRG.

    Google Scholar 

  • Grotschel, M. & M. Padberg [1983]: “Polyhedral Aspects of the Traveling Salesman Problem I: Theory,” to appear in Lawler, E. et al. (eds.): The Traveling Salesman Problem, North-Holland, 1985.

    Google Scholar 

  • Grötschel, M., M. Junger & G. Reinelt (1982a]: “On the Acyclic Subgraph Polytope,” WP 82215-OR, Institut fur Operations Research, Universitat Bonn. (1982).

    Google Scholar 

  • Grötschel, M., M. Jünger & G. Reinelt [1982b]: “Facets of the Linear Ordering Polytope”. WP 82217-OR, Institut fur Operations Research, Universitat Bonn (1982).

    Google Scholar 

  • Grotschel, M., M. Jünger, & G. Reinelt [1983]: “Optimal Triangulation of Large Real World Input-Output Matrices,” Preprint No. 9, Mathematisches Institut, Universitat Augsburg (1983).

    Google Scholar 

  • Held & Karp [1970]: “The Traveling Salesman & Minimum Spanning Trees,” Part 1 Operations Research 18 (1970) 1138–1162, Part II, Mathematical Programming 1 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  • Hong, S., [1972]: “A Linear Programming Approach for the Traveling Salesman Problem,” Ph.D. Thesis, Johns Hopkins University, Baltimore.

    Google Scholar 

  • Hammer, P.L., M. Padberg & U.N. Peled [1975]: “Constraint Pairing in Integer Programming INFOR-Canadian Journal of Operations Research”, 13 (1975) 68–81.

    MATH  MathSciNet  Google Scholar 

  • Johnson, E., & M. Padberg [1981]: “A Note on the Knapsack Problem with Special Ordered Sets” Operations Research Letters, 1 (1981) 38–43.

    Article  MathSciNet  Google Scholar 

  • Johnson, E. & M. Padberg [1982]: “Degree-Two Inequalities, Clique Facets and Biperfect Graphs,” Annals of Discrete Mathematics, 16 (1982) 169–187.

    MATH  MathSciNet  Google Scholar 

  • Knuth, D. [1976]: “The Traveling Salesman Problem,” illustrative example in H. Sullivan: Frontiers of Science, from Microcosm to Macrocosm. The New York Times, February 24 (1976) 18.

    Google Scholar 

  • Krolak, P., Felts, W. & G. Marble, [1971]: “A Man-Machine Approach Toward Solving the Traveling Salesman Problem,” CACM 14 327-334.

    Google Scholar 

  • Land, A., [1979]: “The Solution of 100-City Symmetric Traveling Salesman Problems,” Research Report, London School of Economics, London.

    Google Scholar 

  • Leontief, W. [1936]: “Quantitative Input and Output Relations in the Economic System of the United States,” Review of Economic Systems 18 105-126.

    Google Scholar 

  • Leontief, W., [1951]: The Structure of the American Economy 1919-1939 Oxford University Press, New York.

    Google Scholar 

  • Leontief, W. [1963]: “The Structure of Development” Scientific American, 1963. Leontief, W., [1966]: Input-Output Economics, New York, Oxford University Press.

    Google Scholar 

  • Lin, S. & B. Kerningham, [1973]: “An Effective Heuristic Algorithm for the Traveling Salesman,” Operations Research, 21 498-516.

    Google Scholar 

  • Marsten, R.E. [1981]: “XMP: A Structured Library of Subroutines for Experimental Mathematical Programming,” ACM Transactions on Mathematical Software 7 (1981) 481–497.

    Article  Google Scholar 

  • Marsten, R. E. [1983]: “User’s Guide to IP83,” Technical Report, Dept. of Management Information Systems, University of Arizona.

    Google Scholar 

  • Marsten, R.E. & T.L. Morin, [1978]: “A Hybrid Approach to Discrete Mathematical Programming,” Mathematical Programming 14 (1978) 21–40.

    Article  MATH  MathSciNet  Google Scholar 

  • Martin, R.K. & L. Schräge [1983]: “Subset Coefficient Reduction Cuts for 0/1 Mixed Integer Programming,” Technical Report. Graduate School of Business, University of Chicago, Chicago, Illinois.

    Google Scholar 

  • Milioti8, T., [1976]: “Integer Programming Approaches to the Traveling Salesman Problem,” Mathematical Programming, 10 367-368.

    Google Scholar 

  • Miliotis, T., [1979]: “Using Cutting Planes to Solve the Symmetric Traveling Salesman Problem,” Mathematical Programming, 15 177-188.

    Google Scholar 

  • Padberg, M. [1973]: “On the Facial Structure of Set Packing Polyhedra,” Mathematical Programming, 5 (1973) 199–215.

    Article  MATH  MathSciNet  Google Scholar 

  • Padberg, M. [1974]: “Characterizations of Totally Unimodular, Balanced and Perfect Matrices,” in B. Roy (ed.): Combinatorial Programming: Methods and Applications, D. Reidel Publishing Co., Dordrecht-Holland, 1975 275-284.

    Google Scholar 

  • Padberg, M. [1975]: “A Note on Zero-One Programming,” Operations Research 23(1975) 833-837.

    Google Scholar 

  • Padberg, M. [1977]: “On the Complexity of Set Packing Polyhedra,” Annals of Discrete Mathematics, 1 (1977) 421–434.

    Article  MathSciNet  Google Scholar 

  • Padberg, M. [1979]: “Covering, Packing and Knapsack Problems,” Annals of Discrete Mathematics, 4 (1979) 265–281.

    Article  MATH  MathSciNet  Google Scholar 

  • Padberg, M. & S. Hong, [1980]: “On the Symmetric Traveling Salesman Problem: A Computational Study,” Mathematical Programming Studies, 12( 1980) 78-107.

    Google Scholar 

  • Padberg, M. [1980]: “(l,k)-Configurations and Facets for Packing Problems,” Mathematical Programming 18 (1980) 94–99.

    Article  MATH  MathSciNet  Google Scholar 

  • Padberg, M. & M.R. Rao [1982]: “Odd Minimum Cut-Sets and b-Matchings”, Mathematics of Operations Research, 7 (1982) 67–80.

    Article  MATH  MathSciNet  Google Scholar 

  • Padberg, M. & T. Van Roy, & L. Wolsey, [1982]: “Valid Linear Inequalities for Fixed Charge Problems,” CORE Report No. 8232, Université Catholique, Louvain-la-Neuve, Belgium.

    Google Scholar 

  • Padberg, M. & M. Grötschel [1983]: “Polyhedral Aspects of the Traveling Salesman Problem II: Computation,” to appear in Lawler, E. et al. (eds.): The Traveling Salesman Problem, North-Holland, 1985.

    Google Scholar 

  • Singhai, J. [1982]: “Fixed Order Branch and Bound Methods for Mixed Integer Programming,” Dissertation, Dept. of Management Information Systems, University of Arizona, January, 1982.

    Google Scholar 

  • Thompson, G.L. & R.L. Karp [1964]: “A Heuristic Approach to Solving Traveling Salesman Problems,” Management Science 10 (1964) 225–247.

    Article  Google Scholar 

  • Trotter, L. [1976]: “A Class of Facets for Vertex-Packing Polyhedra,” Discrete Mathematics 12 (1975) 373–388.

    Article  MATH  MathSciNet  Google Scholar 

  • Van Roy, T. & L. Wolsey, [1983]: “Valid Inequalities for Mixed 0-1 Programs,” CORE Report No. 8316, Université Catholique, Louvain-la-Neuve, Belgium.

    Google Scholar 

  • Van Roy, T. & L. Wolsey, [1984a]: “Valid Inequalities and Separation for Uncapacitated Fixed Charge Networks,” CORE Report No* 8410,. Université Catholique Louvain-la-Neuve, Belgium.

    Google Scholar 

  • Van Roy, T. & L. Wolsey, [1984b]: “Solving Mixed Integer Programs by Automatic Reformulation,” CORE Research No. 8432, Université Catholique, Louvain-la-Neuve, Belgium.

    Google Scholar 

  • Weyl, H. [1935]: “Elementare Theorie der konvexen Polyheder,” Comm. Math. Helv., 7 (1935), 290–306 translated in Contributions to the Theory of Games, Vol. I, 3-18, Annals of Mathematics Studies, No. 24, Princeton, 1950.

    Article  MATH  MathSciNet  Google Scholar 

  • Wolsey, L. [1975]: “Faces for a Linear Inequality in 0-1 Variables,” Mathematical Programming 8(1975) 165-178.

    Google Scholar 

  • Young, H.P. [1978]: “On Permutations and Permutation Polytopes,” Mathematical Programming Studies 8 (1978) 128–140.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Center for Applied Mathematics, National Bureau of Standards, Gaithersburg, Maryland, 20899, USA

    Karla Hoffman

  2. Operations Research Division, National Bureau of Standards, Gaithersburg, Maryland, USA

    Karla Hoffman

  3. Graduate School of Business Administration, New York University, New York, 10006, New York, USA

    Manfred Padberg

Authors
  1. Karla Hoffman
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  2. Manfred Padberg
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Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität Stuttgart, Azenbergstraße 12, 7000, Stuttgart 1, Federal Republic of Germany

    Klaus Schittkowski

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© 1985 Springer-Verlag Berlin Heidelberg

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Hoffman, K., Padberg, M. (1985). LP-Based Combinatorial Problem Solving. In: Schittkowski, K. (eds) Computational Mathematical Programming. NATO ASI Series, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82450-0_3

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  • DOI: https://doi.org/10.1007/978-3-642-82450-0_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-82452-4

  • Online ISBN: 978-3-642-82450-0

  • eBook Packages: Springer Book Archive

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