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The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

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Probability on Discrete Structures

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 110))

Abstract

This survey describes a general approach to a class of problems that arise in combinatorial probability and combinatorial optimization. Formally, the method is part of weak convergence theory, but in concrete problems the method has a flavor of its own. A characteristic element of the method is that it often calls for one to introduce a new, infinite, probabilistic object whose local properties inform us about the limiting properties of a sequence of finite problems.

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References

  1. Aldous, D.J. (1990): A random tree model associated with random graphs. Random Structures Algorithms, 1, 383–402.

    Article  MathSciNet  MATH  Google Scholar 

  2. Aldous, D.J. (1991): Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1, 228–266.

    Google Scholar 

  3. Aldous, D.J. (1992): Asymptotics in the random assignment problem. Probab. Th. Rel. Fields, 93, 507–534.

    Google Scholar 

  4. Aldous, D.J. (2001): The C(2) limit in the random assignment problem. Random Structures Algorithms, 18, 381–418.

    Article  MathSciNet  MATH  Google Scholar 

  5. Aldous, D.J. and Bandyopadhyay, A. (2002): A Survey of Max-type Recursive Distributional Equations. Technical Report, U.C. Berkeley.

    Google Scholar 

  6. Aldous, D.J. and Steele, J.M. (1992): Asymptotics of Euclidean Minimal Spanning Trees on Random Samples. Probab. Th. Rel. Fields, 92 247–258.

    Google Scholar 

  7. Aldous, D.J. and Steele, J.M. (2002): The asymptotic essential uniqueness property for minimal spanning trees on random points, manuscript in preparation.

    Google Scholar 

  8. Alexander, K.S. (1995) Percolation and minimal spanning forest in infinite graphs. Ann. Probab. 23, 87–104.

    Google Scholar 

  9. Alm, S.E. and Sorkin, G.B. (2002): Exact expectations and distributions for the random assigmnent problem. Combinatorics, Probability, and Computing 11, 217–248.

    Google Scholar 

  10. Angel, O. and Schramm, O. (2002): Uniform Infinite Planar Triangulations, arXiv: mat h. P R/0207153.

    Google Scholar 

  11. Avis, D., Davis, B., and Steele, J.M. (1988): Probabilistic analysis of a greedy heuristic for Euclidean matching. Probability in the Engineering and Information Sciences, 2, 143–156.

    Article  MATH  Google Scholar 

  12. Avram, F. and Bertsimas, D. (1992): The minimum spanning tree constant in geometric probability and under the independent model: a unified approach. Annals of Applied Probability, 2, 113–130.

    Article  MathSciNet  MATH  Google Scholar 

  13. Balas, E. and Toth, P. (1985): Branch and bound methods. In The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimzation. Lawler, E.L, Lenstra, J.K, Rinnooy Kan, A.H.G., and Smoys, D.B. (eds), Wiley, NY.

    Google Scholar 

  14. Beardwood, J., Halton, J.H., and Hammersley, J.M. (1959): The shortest path through many points. Proceedings of the Cambridge Philosphical Society, 55, 299–327.

    Article  MathSciNet  MATH  Google Scholar 

  15. Benjamini, I and Schramm, O. (2001): Recurrence of distributional limits of finite planar graphs. Electronic Joural of Probability, 6 Paper No. 23, 1–13.

    Google Scholar 

  16. Bezuidenhout, C., Grimmett, G., and Löffler, A. (1998): Percolation and minimal spanning trees. J. Statist. Phys., 92, 1–34.

    Article  MathSciNet  MATH  Google Scholar 

  17. Bollobas, B. (1985): Random Graphs. Academic Press, London.

    MATH  Google Scholar 

  18. Devroye, L. (1998): Branching processes and their application in the analysis of tree structures and tree algorithms In M. Habib, editor, Probabilistic Methods for Algorithmic Discrete Mathematics, Springer-Verlag.

    Google Scholar 

  19. Dyer, M. E., Frieze, A. M., and McDiarmid, C. (1986): On linear programs with random costs. Math. Programming, 35, 3–16.

    Article  MathSciNet  MATH  Google Scholar 

  20. Eppstein, D., Paterson, M.S. and Yao, F.F. (1997): On nearest-neighbor graphs. Discrete Comput. Geom., 17, 263–282.

    MathSciNet  MATH  Google Scholar 

  21. Frieze, A.M. (1985): On the value of a random minimum spanning tree problem. Discrete Appl. Math., 10, 47–56.

    MathSciNet  MATH  Google Scholar 

  22. Frieze, A.M. and McDiarmid, C.J.H. (1989): On random minimum lenght spanning trees. Combinatorica, 9, 363–374.

    Article  MathSciNet  MATH  Google Scholar 

  23. Frieze, A. and Sorkin, G.B. (2001): The probabilistic relationship between the assignment problem and asymmetric traveling salesman problems. Proceedings of SODA, ACM Publishers, 652–660.

    Google Scholar 

  24. Frieze, A. and Suen, S. (1994): On the independence number of random cubic graphs. Random Structures and Algorithms, 5, 640–664.

    MathSciNet  Google Scholar 

  25. Gamarnik, D. (2002): Linear Phase Transition in Random Linear Constraint Satisfaction Problem. Technical Report: IBM T.J. Watson Research Center.

    Google Scholar 

  26. Grimmett, G.R. (1980): Random labelled trees and their branching networks. J. Austral. Math. Soc. (Ser. A), 30, 229–237.

    Article  MathSciNet  MATH  Google Scholar 

  27. Füredi, Z. (1995): The expected size of a random sphere of influence graph, Intuitive Geometry, Bolyai Mathematical Society, 6, 319–326.

    Google Scholar 

  28. Hara, T. and Slade, G. (1990): Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys., 128, 333–391.

    Article  MathSciNet  MATH  Google Scholar 

  29. Hara, T. and Slade, G. (1994): Mean-field behaviour and the lace expansion. Probability and Phase Transition (G. Grimmett, ed.), Kluwer, Dordrecht.

    Google Scholar 

  30. Harris, T.H. (1989): The Theory of Branching Processes. Dover Publications, New York.

    Google Scholar 

  31. Hartmann, A.K. and Weigt, M. (2001): Statistical mechanics perspective on the phase transition in vertex covering of finite-connectivity random graphs. Theoretical Computer Science, 265, 199–225.

    Article  MathSciNet  MATH  Google Scholar 

  32. Hayen, A. and Quine, M.P. (2000): The proportion of triangles in a PoissonVoronoi tessellation of the plane. Adv. in Appl. Probab., 32, 67–74.

    Google Scholar 

  33. Hitczenko, P., Janson, S., and Yukich, J.E. (1999): On the variance of the random sphere of influence graph. Random Structures Algorithms, 14, 139–152.

    Article  MathSciNet  MATH  Google Scholar 

  34. Henze, N. (1987): On the fraction of random points with specified nearest neighbor interrelations and degree of attraction. Adv. Appl. Prob., 19, 873–895.

    Article  MathSciNet  MATH  Google Scholar 

  35. Hochbaum, D. and Steele, J.M. (1982): Steinhaus’ geometric location problem for random samples in the plane. Advances in Applied Probability, 14, 55–67.

    Article  MathSciNet  Google Scholar 

  36. Janson, S., Luczak, T., and Rucinski (2000): Random Graphs. Wiley Inter-science Publishers, New York.

    Google Scholar 

  37. Karp, R.M. (1979): A patching algorithm for the non-symmetric traveling salesman problem. SIAM Journal on Computing, 8, 561–573.

    Article  MathSciNet  MATH  Google Scholar 

  38. Karp, R.M. (1987): An upper bound on the expected cost of an optimal assignment. In: Discrete Algorithms and Complexity: Proceedings of the Japan-U. S. Joint Seminar, Academic Press, New York.

    Google Scholar 

  39. Karp, R.M. and Steele, J.M. (1985): Probabilistic analysis of heuristics. In The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimzation. Lawler, E.L, Lenstra, J.K, Rinnooy Kan, A.H.G., and Smoys, D.B. ( eds ), Wiley, NY, 181–205.

    Google Scholar 

  40. Kesten, H. and Lee, S. (1996): The central limit theorem for weighted minimal spanning trees on random points. Annals of Applied Probability, 6, 495–527.

    Article  MathSciNet  MATH  Google Scholar 

  41. Koebe, P. (1936): Kontaktprobleme der Konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88, 141–164.

    Google Scholar 

  42. Kurtzberg, J. M. (1962): On approximation methods for the assignment problem. J. Assoc. Comput. Mach., 9, 419–439.

    Article  MathSciNet  MATH  Google Scholar 

  43. Lee, S. (1997): The central limit theorem for Euclidean minimal spanning trees I. Annals of Applied Probability, 7, 996–1020.

    Article  MathSciNet  MATH  Google Scholar 

  44. Lee, S. (1999): The central limit theorem for Euclidean minimal spanning trees II., Advances in Applied Probability, 31, 969–984.

    Article  MathSciNet  MATH  Google Scholar 

  45. Lee, S. and Su, Z. (2002): On the fluctutation in the random assigmnent problem. Commun. Korean. Math. Soc., 17, 321–330.

    Article  MathSciNet  MATH  Google Scholar 

  46. Linusson, S. and Wästlund, J. (2003): A Proof of Parisi’s Conjecture on the Random Assignment Problem. Unpublished.

    Google Scholar 

  47. Lovâsz, L. and Plummer, M.D. (1986): Matching Theory. Annals of Discrete Mathematics, vol. 29. North-Holland Publishers. Amsterdam.

    Google Scholar 

  48. Mahmoud, H.M. (1992): Evolution of Random Search Trees, Wiley, New York, 1992.

    MATH  Google Scholar 

  49. Matousek, J. and Nestr“il, Y. (1998): Discrete Mathematics, Oxford Universtity Press, Oxford.

    Google Scholar 

  50. Meyn, S.P. and Tweedie, R.L. (1993): Markov Chains and Stochastic Stability, Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  51. Mézard, M. and Parisi, G. (1987): On the solution of the random link matching problem. J. Physique, 48, 1451–1459.

    Article  Google Scholar 

  52. Nair, C. and Prabhakar, B. and Sharma, M. (2003): A Proof of Parisi’s Conjecture for the Finite Random Assignment Problem. Unpublished.

    Google Scholar 

  53. Parisi, G. (1998): A conjecture on random bipartite matching. ArXiv Condmat 13, 277–303.

    Google Scholar 

  54. Penrose, M.D. (1996): The random minimal spanning tree in high dimensions. Ann of Probability, 24 1903–1925.

    Article  MathSciNet  MATH  Google Scholar 

  55. Penrose, M.D. and Yukich, J.E. (2002): Weak laws of large numbers geometric probability, Annals of Applied Probability, 13, 277–303.

    MathSciNet  Google Scholar 

  56. Propp, J. and Wilson D. (1998): Coupling from the past: a user’s guide, In D. Aldous and J. Propp, editors, Microsurveys in Discrete Probability, number 41 in DIMACS Ser. Discrete Math. Theoret. Comp. Sci., pages 181–192.

    Google Scholar 

  57. Steele, J.M. (1981): Subadditive Euclidean functionals and non-linear growth in geometric probability. Annals of Probability, 9, 365–376.

    Article  MathSciNet  MATH  Google Scholar 

  58. Steele, J.M. (1982): Optimal triangulation of random samples in the plane. Annals of Probability, 10, 548–553.

    Article  MathSciNet  MATH  Google Scholar 

  59. Steele, J.M. (1987): On Frieze’s ((3) limit for the lenths of minimal spanning trees, Discrete Applied Mathematics, 18, 99–103.

    Article  MathSciNet  MATH  Google Scholar 

  60. Steele, J.M. (1988): Growth rates of Euclidean minimal spanning trees with power weighted edges. Annals of Probability, 16, 1767–1787, 1988.

    MathSciNet  Google Scholar 

  61. Steele, J.M. (1992): Euclidean semi-matchings of random samples. Mathematical Programming, 53, 127–146, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  62. Steele, J.M. (1997): Probability Theory and Combinatorial Optimization, NSFCBMS Volume 69. Society for Industrial and Applied Mathematics, Philadelphia.

    Book  Google Scholar 

  63. Steele, J.M., Shepp, L.A. J.M. Eddy, W. (1987): On the number of leaves of a Euclidean minimal spanning tree. J. Appl. Probab., 24, 809–826.

    MathSciNet  MATH  Google Scholar 

  64. Talagrand, M. (1995): Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. IHES, 81, 73–205.

    Google Scholar 

  65. Vitter, J.S. and Flajolet, P. (1990): Analysis of algorithms and data structures. In Handbook of Theoretical Computer Science, volume A: Algorithms and Complexity (Chapter 9), North-Holland, 431–524.

    Google Scholar 

  66. Walkup, D. W. (1979): On the expected value of a random assigmnent problem. SIAM J. Comput., 8, 440–442.

    Google Scholar 

  67. Yukich, J.E. (1998): Probability Theory of Classical Euclidean Optimization Problems, Lecture Notes in Mathematics, 1675, Springer-Verlag, New York.

    Google Scholar 

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Authors
  1. David Aldous
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  2. J. Michael Steele
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Editor information

Editors and Affiliations

  1. Cornell University Department of Mathematics, Malott Hall 310, 14853-4201, Ithaca, NY, USA

    Harry Kesten

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Aldous, D., Steele, J.M. (2004). The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence. In: Kesten, H. (eds) Probability on Discrete Structures. Encyclopaedia of Mathematical Sciences, vol 110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09444-0_1

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  • DOI: https://doi.org/10.1007/978-3-662-09444-0_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-05647-5

  • Online ISBN: 978-3-662-09444-0

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