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The minimum perfect matching in pseudo-dimension 0 < q < 1

Published online by Cambridge University Press:  27 October 2020

Joel Larsson
Affiliation:
Department of Statistics, Lund University, Lund, Scania, Sweden
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Abstract

It is known that for Kn,n equipped with i.i.d. exp (1) edge costs, the minimum total cost of a perfect matching converges to $\zeta(2)=\pi^2/6$ in probability. Similar convergence has been established for all edge cost distributions of pseudo-dimension $q \geq 1$ . In this paper we extend those results to all real positive q, confirming the Mézard–Parisi conjecture in the last remaining applicable case.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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© The Author(s), 2020. Published by Cambridge University Press

References

Aldous, D. J. (1992) Asymptotics in the random assignment problem. Probab. Theory Rel. Fields 93 507534.CrossRefGoogle Scholar
Aldous, D. J. (2001) The $\zeta(2)$ limit in the random assignment problem. Random Struct. Algorithms 18 381418.CrossRefGoogle Scholar
Aldous, D. and Steele, J. M. (2004) The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures (Kesten, H., ed.), Vol. 110 of Encyclopaedia of Mathematical Sciences, pp. 172. Springer.CrossRefGoogle Scholar
Karp, R. M. (1987) An upper bound on the expected cost of an optimal assignment. In Discrete Algorithms and Complexity, Vol. 15 of Perspectives in Computing, pp. 14. Academic Press.CrossRefGoogle Scholar
Krokhmal, P. A. and Pardalos, P. M. (2009) Random assignment problems. European J. Oper. Res. 194 117.CrossRefGoogle Scholar
Linusson, S. and Wästlund, J. (2004) A proof of Parisi’s conjecture on the random assignment problem. Probab. Theory Relat. Fields 128 419440.CrossRefGoogle Scholar
Lovász, L. (2012) Large Networks and Graph Limits, Vol. 60 of Colloquium Publications. American Mathematical Society.CrossRefGoogle Scholar
Mézard, M. and Parisi, G. (1985) Replicas and optimization. J. Physique Lett. 46 L771L778.CrossRefGoogle Scholar
Nair, C., Prabhakar, B. and Sharma, M. (2005) Proofs of the Parisi and Coppersmith–Sorkin random assignment conjectures. Random Struct. Algorithms 27 413443.CrossRefGoogle Scholar
Parisi, G. (1998) A conjecture on random bipartite matching. arXiv:cond-mat/9801176v1Google Scholar
Salez, J. and Shah, D. (2009) Belief propagation: an asymptotically optimal algorithm for the random assignment problem. Math. Oper. Res. 34 468480.CrossRefGoogle Scholar
Walkup, D. W. (1979) On the expected value of a random assignment problem. SIAM J. Comput. 8 440442.CrossRefGoogle Scholar
Wästlund, J. (2005) A proof of a conjecture of Buck, Chan, and Robbins on the expected value of the minimum assignment. Random Struct. Algorithms 26 237251.CrossRefGoogle Scholar
Wästlund, J. (2012) Replica symmetry of the minimum matching. Ann. of Math. 175 10611091.CrossRefGoogle Scholar

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