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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aldous, D.J. (1990): A random tree model associated with random graphs. Random Structures Algorithms, 1, 383–402.
Aldous, D.J. (1991): Asymptotic fringe distributions for general families of random trees. Ann. Appl. Probab. 1, 228–266.
Aldous, D.J. (1992): Asymptotics in the random assignment problem. Probab. Th. Rel. Fields, 93, 507–534.
Aldous, D.J. (2001): The C(2) limit in the random assignment problem. Random Structures Algorithms, 18, 381–418.
Aldous, D.J. and Bandyopadhyay, A. (2002): A Survey of Max-type Recursive Distributional Equations. Technical Report, U.C. Berkeley.
Aldous, D.J. and Steele, J.M. (1992): Asymptotics of Euclidean Minimal Spanning Trees on Random Samples. Probab. Th. Rel. Fields, 92 247–258.
Aldous, D.J. and Steele, J.M. (2002): The asymptotic essential uniqueness property for minimal spanning trees on random points, manuscript in preparation.
Alexander, K.S. (1995) Percolation and minimal spanning forest in infinite graphs. Ann. Probab. 23, 87–104.
Alm, S.E. and Sorkin, G.B. (2002): Exact expectations and distributions for the random assigmnent problem. Combinatorics, Probability, and Computing 11, 217–248.
Angel, O. and Schramm, O. (2002): Uniform Infinite Planar Triangulations, arXiv: mat h. P R/0207153.
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.
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.
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.
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.
Benjamini, I and Schramm, O. (2001): Recurrence of distributional limits of finite planar graphs. Electronic Joural of Probability, 6 Paper No. 23, 1–13.
Bezuidenhout, C., Grimmett, G., and Löffler, A. (1998): Percolation and minimal spanning trees. J. Statist. Phys., 92, 1–34.
Bollobas, B. (1985): Random Graphs. Academic Press, London.
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.
Dyer, M. E., Frieze, A. M., and McDiarmid, C. (1986): On linear programs with random costs. Math. Programming, 35, 3–16.
Eppstein, D., Paterson, M.S. and Yao, F.F. (1997): On nearest-neighbor graphs. Discrete Comput. Geom., 17, 263–282.
Frieze, A.M. (1985): On the value of a random minimum spanning tree problem. Discrete Appl. Math., 10, 47–56.
Frieze, A.M. and McDiarmid, C.J.H. (1989): On random minimum lenght spanning trees. Combinatorica, 9, 363–374.
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.
Frieze, A. and Suen, S. (1994): On the independence number of random cubic graphs. Random Structures and Algorithms, 5, 640–664.
Gamarnik, D. (2002): Linear Phase Transition in Random Linear Constraint Satisfaction Problem. Technical Report: IBM T.J. Watson Research Center.
Grimmett, G.R. (1980): Random labelled trees and their branching networks. J. Austral. Math. Soc. (Ser. A), 30, 229–237.
Füredi, Z. (1995): The expected size of a random sphere of influence graph, Intuitive Geometry, Bolyai Mathematical Society, 6, 319–326.
Hara, T. and Slade, G. (1990): Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys., 128, 333–391.
Hara, T. and Slade, G. (1994): Mean-field behaviour and the lace expansion. Probability and Phase Transition (G. Grimmett, ed.), Kluwer, Dordrecht.
Harris, T.H. (1989): The Theory of Branching Processes. Dover Publications, New York.
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.
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.
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.
Henze, N. (1987): On the fraction of random points with specified nearest neighbor interrelations and degree of attraction. Adv. Appl. Prob., 19, 873–895.
Hochbaum, D. and Steele, J.M. (1982): Steinhaus’ geometric location problem for random samples in the plane. Advances in Applied Probability, 14, 55–67.
Janson, S., Luczak, T., and Rucinski (2000): Random Graphs. Wiley Inter-science Publishers, New York.
Karp, R.M. (1979): A patching algorithm for the non-symmetric traveling salesman problem. SIAM Journal on Computing, 8, 561–573.
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.
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.
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.
Koebe, P. (1936): Kontaktprobleme der Konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88, 141–164.
Kurtzberg, J. M. (1962): On approximation methods for the assignment problem. J. Assoc. Comput. Mach., 9, 419–439.
Lee, S. (1997): The central limit theorem for Euclidean minimal spanning trees I. Annals of Applied Probability, 7, 996–1020.
Lee, S. (1999): The central limit theorem for Euclidean minimal spanning trees II., Advances in Applied Probability, 31, 969–984.
Lee, S. and Su, Z. (2002): On the fluctutation in the random assigmnent problem. Commun. Korean. Math. Soc., 17, 321–330.
Linusson, S. and Wästlund, J. (2003): A Proof of Parisi’s Conjecture on the Random Assignment Problem. Unpublished.
Lovâsz, L. and Plummer, M.D. (1986): Matching Theory. Annals of Discrete Mathematics, vol. 29. North-Holland Publishers. Amsterdam.
Mahmoud, H.M. (1992): Evolution of Random Search Trees, Wiley, New York, 1992.
Matousek, J. and Nestr“il, Y. (1998): Discrete Mathematics, Oxford Universtity Press, Oxford.
Meyn, S.P. and Tweedie, R.L. (1993): Markov Chains and Stochastic Stability, Springer-Verlag, New York.
Mézard, M. and Parisi, G. (1987): On the solution of the random link matching problem. J. Physique, 48, 1451–1459.
Nair, C. and Prabhakar, B. and Sharma, M. (2003): A Proof of Parisi’s Conjecture for the Finite Random Assignment Problem. Unpublished.
Parisi, G. (1998): A conjecture on random bipartite matching. ArXiv Condmat 13, 277–303.
Penrose, M.D. (1996): The random minimal spanning tree in high dimensions. Ann of Probability, 24 1903–1925.
Penrose, M.D. and Yukich, J.E. (2002): Weak laws of large numbers geometric probability, Annals of Applied Probability, 13, 277–303.
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.
Steele, J.M. (1981): Subadditive Euclidean functionals and non-linear growth in geometric probability. Annals of Probability, 9, 365–376.
Steele, J.M. (1982): Optimal triangulation of random samples in the plane. Annals of Probability, 10, 548–553.
Steele, J.M. (1987): On Frieze’s ((3) limit for the lenths of minimal spanning trees, Discrete Applied Mathematics, 18, 99–103.
Steele, J.M. (1988): Growth rates of Euclidean minimal spanning trees with power weighted edges. Annals of Probability, 16, 1767–1787, 1988.
Steele, J.M. (1992): Euclidean semi-matchings of random samples. Mathematical Programming, 53, 127–146, 1992.
Steele, J.M. (1997): Probability Theory and Combinatorial Optimization, NSFCBMS Volume 69. Society for Industrial and Applied Mathematics, Philadelphia.
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.
Talagrand, M. (1995): Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. IHES, 81, 73–205.
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.
Walkup, D. W. (1979): On the expected value of a random assigmnent problem. SIAM J. Comput., 8, 440–442.
Yukich, J.E. (1998): Probability Theory of Classical Euclidean Optimization Problems, Lecture Notes in Mathematics, 1675, Springer-Verlag, New York.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
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
eBook Packages: Springer Book Archive