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Anomalous Scaling of the Optimal Cost in the One-Dimensional Random Assignment Problem

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Abstract

We consider the random Euclidean assignment problem on the line between two sets of N random points, independently generated with the same probability density function \(\varrho \). The cost of the matching is supposed to be dependent on a power \(p>1\) of the Euclidean distance of the matched pairs. We discuss an integral expression for the average optimal cost for \(N\gg 1\) that generalizes a previous result obtained for \(p=2\). We also study the possible divergence of the given expression due to the vanishing of the probability density function. The provided regularization recipe allows us to recover the proper scaling law for the cost in the divergent cases, and possibly some of the involved coefficients. The possibility that the support of \(\varrho \) is a disconnected interval is also analysed. We exemplify the proposed procedure and we compare our predictions with the results of numerical simulations.

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Notes

  1. Here and in the following \(\theta (x)\) is the Heaviside function, such that \(\theta (x)=1\) if \(x\ge 0\), \(\theta (x)=0\) otherwise.

  2. To obtain Eq. (17) we have introduced the Gauss hypergeometric function

    $$\begin{aligned} {}_2F_1[a,b;c;z]{:}{=}\sum _{k=0}^\infty \frac{(a)_k(b)_k}{(c)_k}\frac{z^k}{k!},\quad (x)_k{:}{=}\prod _{n=0}^{k-1}(x+n), \end{aligned}$$

    and we have used the fact that

    $$\begin{aligned} \sum _{k=1}^N\left( {\begin{array}{c}N\\ k\end{array}}\right) ^2 k^2z^{k}=N^2z\sum _{k=0}^\infty \frac{(1-N)_k(1-N)_k}{(1)_k}\frac{z^k}{k!}=N^2z\,{}_2F_1\left[ 1-N,1-N;1;z\right] . \end{aligned}$$

References

  1. Papadimitriou, C., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover Books on Computer Science Series. Dover Publications, New York (1998)

    Google Scholar 

  2. Ajtai, M., Komlós, J., Tusnády, G.: On optimal matchings. Combinatorica 4, 259–264 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Caracciolo, S., Lucibello, C., Parisi, G., Sicuro, G.: Scaling hypothesis for the Euclidean bipartite matching problem. Phys. Rev. E 90, 012118 (2014)

    Article  ADS  Google Scholar 

  4. Caracciolo, S., Sicuro, G.: Quadratic stochastic Euclidean bipartite matching problem. Phys. Rev. Lett. 115, 230601 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  5. Ambrosio, L., Stra, F., Trevisan, D.: A PDE approach to a 2-dimensional matching problem. Prob. Theory Relat. Fields (2018). https://doi.org/10.1007/s00440-018-0837-x

  6. Ledoux, M.: On optimal matching of Gaussian samples. Zap. Nauchn. Sem. POMI 457, 226–264 (2017)

    MathSciNet  Google Scholar 

  7. Talagrand, M.: Scaling and non-standard matching theorems. C. R. Acad. Sci. Paris. Ser. I 356, 692–695 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. Lecture Notes in Physics Series. World Scientific Publishing Company, Inc., Teaneck (1987)

    MATH  Google Scholar 

  9. Orland, H.: Mean-field theory for optimization problems. J. Phys. Lett. 46, 763–770 (1985)

    Article  Google Scholar 

  10. Mézard, M., Parisi, G.: Replicas and optimization. J. Phys. Lett. 46, 771–778 (1985)

    Article  Google Scholar 

  11. Mézard, M., Parisi, G.: On the solution of the random link matching problems. J. Phys. 48, 1451–1459 (1987)

    Article  Google Scholar 

  12. Aldous, D.J.: Asymptotics in the random assignment problem. Probab. Theory Relat. Fields 93, 507–534 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Linusson, S., Wästlund, J.: A proof of Parisi’s conjecture on the random assignment problem. Probab. Theory Relat. Fields 128, 419–440 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nair, C., Prabhakar, B., Sharma, M.: Proofs of the Parisi and Coppersmith–Sorkin random assignment conjectures. Random Struct. Algorithms 27, 413–444 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Parisi, G., Ratiéville, M.: On the finite size corrections to some random matching problems. Eur. Phys. J. B 29, 457–468 (2002)

    Article  ADS  Google Scholar 

  16. Mézard, M., Parisi, G.: The Euclidean matching problem. J. Phys. 49, 2019–2025 (1988)

    Article  MathSciNet  Google Scholar 

  17. Houdayer, J., Boutet de Monvel, J.H., Martin, O.C.: Comparing mean field and Euclidean matching problems. Eur. Phys. J. B 6, 383–393 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  18. Lucibello, C., Parisi, G., Sicuro, G.: One-loop diagrams in the random Euclidean matching problem. Phys. Rev. E 95(1), 012302 (2017)

    Article  ADS  Google Scholar 

  19. Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002)

    Article  ADS  Google Scholar 

  20. Braunstein, A., Mézard, M., Zecchina, R.: Survey propagation: an algorithm for satisfiability, Random Struct. Algorithms 27, 201–226 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mézard, M., Montanari, A.: Information, Physics, and Computation. Oxford Graduate Texts. OUP, Oxford (2009)

    Book  MATH  Google Scholar 

  22. Villani, C.: Optimal Transport. Old and New. Springer, New York (2009)

    Book  MATH  Google Scholar 

  23. Boniolo, E., Caracciolo, S., Sportiello, A.: Correlation function for the Grid-Poisson Euclidean matching on a line and on a circle. J. Stat. Mech. Theory Exp. 2014, P11023 (2014)

    Article  MathSciNet  Google Scholar 

  24. Caracciolo, S., D’Achille, M., Sicuro, G.: Random Euclidean matching problems in one dimension. Phys. Rev. E 96(4), 042102 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  25. Caracciolo, S., Sicuro, G.: One-dimensional Euclidean matching problem: exact solutions, correlation functions, and universality. Phys. Rev. E 90, 042112 (2014)

    Article  ADS  Google Scholar 

  26. Del Barrio, E., Giné, E., Utzet, F.: Asymptotics for \(L_2\) functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances. Bernoulli 11, 131–189 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. McCann, R.J.: Exact solutions to the transportation problem on the line. Proc. R. Soc. A Math. Phys. Eng. Sci. 455, 1341–1380 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Bobkov, S., Ledoux, M.: Preprint. https://perso.math.univ-toulouse.fr/ledoux/files/2016/12/MEMO.pdf (2016)

  29. Caracciolo, S., Sicuro, G.: Scaling hypothesis for the Euclidean bipartite matching problem. II. Correlation functions. Phys. Rev. E 91, 062125 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  30. Caracciolo, S., Di Gioacchino, A., Malatesta, E.M.: Plastic number and possible optimal solutions for an Euclidean 2-matching in one dimension. J. Stat. Mech. Theory Exp. 2018, 083402 (2018)

    Article  MathSciNet  Google Scholar 

  31. Caracciolo, S., Di Gioacchino, A., Gherardi, M., Malatesta, E.M.: Solution for a bipartite Euclidean traveling-salesman problem in one dimension. Phys. Rev. E 97(5), 052109 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  32. Caracciolo, S., Di Gioacchino, A., Malatesta, E M., Vanoni, C.: Average optimal cost for the Euclidean TSP in one dimension. ArXiv e-prints (Preprint 1811.08265) (2018)

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Acknowledgements

The authors are grateful to Luigi Ambrosio for useful discussions and correspondence, and for pointing us to Ref. [28]. The authors also thank Giorgio Parisi, for many discussions on the assignment problem in Euclidean spaces. GS acknowledges the financial support of the Simons foundation (Grant No. 454949, Giorgio Parisi). The work presented in this article was supported by the project “Meccanica statistica e complessità”, a research grant funded by PRIN 2015 (Agreement no. 2015K7KK8L).

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Authors and Affiliations

  1. University of Milan and INFN, via Celoria 16, 20133, Milan, Italy

    Sergio Caracciolo

  2. Centre CEA de Saclay, Gif-sur-Yvette, France

    Matteo D’Achille

  3. CIRB Collège de France, 11 Place Marcelin Berthelot, 75231, Paris, France

    Matteo D’Achille

  4. LI-PaRAD - Université de Versailles Saint-Quentin-en-Yvelines, Versailles, France

    Matteo D’Achille

  5. Université Paris Saclay, Paris, France

    Matteo D’Achille

  6. Sapienza Università di Roma, P.le A. Moro 2, 00185, Rome, Italy

    Gabriele Sicuro

Authors
  1. Sergio Caracciolo
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  2. Matteo D’Achille
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  3. Gabriele Sicuro
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Correspondence to Gabriele Sicuro.

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Caracciolo, S., D’Achille, M. & Sicuro, G. Anomalous Scaling of the Optimal Cost in the One-Dimensional Random Assignment Problem. J Stat Phys 174, 846–864 (2019). https://doi.org/10.1007/s10955-018-2212-9

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