The ζ(2) limit in the random assignment problem

The random assignment (or bipartite matching) problem asks about An=minπ ∑  i=1nc(i, π(i)), where (c(i, j)) is a n×n matrix with i.i.d. entries, say with exponential(1) distribution, and the minimum is over permutations π. Mézard and Parisi (1987) used the replica method from statistical physics to argue nonrigorously that EAn→ζ(2)=π2/6. Aldous (1992) identified the limit in terms of a matching problem on a limit infinite tree. Here we construct the optimal matching on the infinite tree. This yields a rigorous proof of the ζ(2) limit and of the conjectured limit distribution of edge‐costs and their rank‐orders in the optimal matching. It also yields the asymptotic essential uniqueness property: every almost‐optimal matching coincides with the optimal matching except on a small proportion of edges. ©2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 381–418, 2001

[1]  Constance de Koning,et al.  Editors , 2003, Annals of Emergency Medicine.

[2]  Michel Talagrand.  An assignment problem at high temperature , 2003 .

[3]  Sven Erick Alm,et al.  Exact Expectations And Distributions For The Random Assignment Problem , 2002, Comb. Probab. Comput..

[4]  D. P. Robbins,et al.  On the expected value of the minimum assignment , 2000, Random Struct. Algorithms.

[5]  Svante Linusson,et al.  A Generalization of the Random Assignment Problem , 2000, math/0006146.

[6]  D. Aldous.  The percolation process on a tree where infinite clusters are frozen , 2000, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  V. Dotsenko.  Exact solution of the random bipartite matching model , 1999, cond-mat/9911477.

[8]  D. Coppersmith,et al.  Constructive bounds and exact expectation for the random assignment problem , 1999 .

[9]  Don Coppersmith,et al.  Constructive bounds and exact expectations for the random assignment problem , 1998, Random Struct. Algorithms.

[10]  O. Martin,et al.  The Stochastic Traveling Salesman Problem: Finite Size Scaling and the Cavity Prediction , 1998, cond-mat/9802295.

[11]  O. Martin,et al.  Comparing mean field and Euclidean matching problems , 1998, cond-mat/9803195.

[12]  G. Parisi.  A Conjecture on random bipartite matching , 1998, cond-mat/9801176.

[13]  M. Talagrand.  Huge random structures and mean field models for spin glasses. , 1998 .

[14]  S. Rachev,et al.  Probability metrics and recursive algorithms , 1995, Advances in Applied Probability.

[15]  Andrew J. Lazarus,et al.  Certain expected values in the random assignment problem , 1993, Oper. Res. Lett..

[16]  Michel X. Goemans,et al.  A Lower Bound on the Expected Cost of an Optimal Assignment , 1993, Math. Oper. Res..

[17]  D. Aldous.  Asymptotics in the random assignment problem , 1992 .

[18]  G. Parisi.  Field Theory, Disorder and Simulations , 1992 .

[19]  David Aldous,et al.  A Random Tree Model Associated with Random Graphs , 1990, Random Struct. Algorithms.

[20]  W. Krauth,et al.  The Cavity Method and the Travelling-Salesman Problem , 1989 .

[21]  M. Mézard,et al.  On the solution of the random link matching problems , 1987 .

[22]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[23]  R. Karp.  An Upper Bound on the Expected Cost of an Optimal Assignment , 1987 .

[24]  J. Steele.  Probability theory and combinatorial optimization , 1987 .

[25]  M. Mézard,et al.  A replica analysis of the travelling salesman problem , 1986 .

[26]  Martin E. Dyer,et al.  On linear programs with random costs , 1986, Math. Program..

[27]  M. Mézard,et al.  Replicas and optimization , 1985 .

[28]  Alan M. Frieze,et al.  On the value of a random minimum spanning tree problem , 1985, Discret. Appl. Math..

[29]  Richard M. Karp,et al.  Maximum Matchings in Sparse Random Graphs , 1981, FOCS 1981.

[30]  Richard M. Karp,et al.  A Patching Algorithm for the Nonsymmetric Traveling-Salesman Problem , 1979, SIAM J. Comput..

[31]  David W. Walkup,et al.  On the Expected Value of a Random Assignment Problem , 1979, SIAM J. Comput..

[32]  D. Bailin.  Field theory , 1979, Nature.

[33]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.