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Sequential Importance Sampling for Estimating the Number of Perfect Matchings in Bipartite Graphs: An Ongoing Conversation with Laci

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Book cover Building Bridges II

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 28))

Abstract

Sequential importance sampling offers an alternative way to approximately evaluate the permanent. It is a stochastic algorithm which seems to work in practice but has eluded analysis. This paper offers examples where the analysis can be carried out and the first general bounds for the sample size required. This uses a novel importance sampling proof of Brégman’s inequality due to Lovász.

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References

  1. Blitzstein J, Diaconis P (2010) A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math 6(4):489–522, https://doi.org/10.1080/15427951.2010.557277

    Article  MathSciNet  MATH  Google Scholar 

  2. Chatterjee S, Diaconis P (2018) The sample size required in importance sampling. Ann Appl Probab 28(2):1099–1135, https://doi.org/10.1214/17-AAP1326

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen Y, Diaconis P, Holmes SP, Liu JS (2005) Sequential Monte Carlo methods for statistical analysis of tables. J Amer Statist Assoc 100(469):109–120

    Article  MathSciNet  Google Scholar 

  4. Chung F, Diaconis P, Graham R (2018) Permanental generating functions and sequential importance sampling, Adv. Applied Mathematics to appear

    Google Scholar 

  5. Diaconis P, Kolesnik B (2018) Randomized sequential importance sampling for estimating the number of perfect matchings in bipartite graphs, arxiv:1907.02333

  6. Diaconis P, Graham R, Holmes SP (2001) Statistical problems involving permutations with restricted positions. In: State of the art in probability and statistics (Leiden, 1999), IMS Lecture Notes Monogr. Ser., vol 36, Inst. Math. Statist., Beachwood, OH, pp 195–222

    Google Scholar 

  7. Dyer M, Jerrum M, Müller H (2017) On the switch Markov chain for perfect matchings. J ACM 64(2):Art. 12, 33, https://doi.org/10.1145/2822322

    Article  MathSciNet  Google Scholar 

  8. Fox J, He X, Manners F (2017) A proof of Tomescu’s graph coloring conjecture. ArXiv e-prints arxiv:1712.06067

  9. Jerrum M, Sinclair A, Vigoda E (2004) A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J ACM 51(4):671–697 (electronic)

    Article  MathSciNet  Google Scholar 

  10. Knuth DE (1976) Mathematics and computer science: Coping with finiteness. Science 194(4271):1235–1242, https://doi.org/10.1126/science.194.4271.1235

    Article  MathSciNet  MATH  Google Scholar 

  11. Knuth DE (2018) The Art of Computer Programming, Volume 4B, 1st edn. Addison-Wesley Professional, Fascicle 5: Mathematical Preliminaries Redux; Backtracking; Dancing Links

    Google Scholar 

  12. Levin DA, Peres Y (2017) Counting walks and graph homomorphisms via Markov chains and importance sampling. Amer Math Mon 124(7):637–641, https://doi.org/10.4169/amer.math.monthly.124.7.637

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu JS (2001) Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics, Springer-Verlag, New York

    Google Scholar 

  14. Lovász L, Plummer MD (2009) Matching Theory. AMS Chelsea Publishing, Providence, RI, https://doi.org/10.1090/chel/367

  15. Spencer J (1990) The probabilistic lens: Sperner, Turán and Brégman revisited. In: A Tribute to Paul Erdős, Cambridge Univ. Press, Cambridge, pp 391–396

    Chapter  Google Scholar 

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Acknowledgements

Support is acknowledged by National Science Foundation award DMS 1608182. My thanks to Paulo Ornstein for trying these estimators out; to Sourav Chatterjee for help with importance sampling; and to Joe Blitzstein whose thesis work suggested using sequential importance sampling for bipartite matchings, Leo Nagami and Don Knuth who gave detailed corrections added in proof. Thanks as well to Fan Chung, Ron Graham, and Brett Kolesnik, and to Laci Lovász for starting the conversation.

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Authors and Affiliations

  1. Departments of Mathematics and Statistics, Stanford University, Sequoia Hall, 390 Serra Mall, Stanford, CA, 94305, USA

    Persi Diaconis

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  1. Persi Diaconis
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Correspondence to Persi Diaconis .

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Editors and Affiliations

  1. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  2. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gyula O. H. Katona

  3. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Attila Sali

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© 2019 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature

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Diaconis, P. (2019). Sequential Importance Sampling for Estimating the Number of Perfect Matchings in Bipartite Graphs: An Ongoing Conversation with Laci. In: Bárány, I., Katona, G., Sali, A. (eds) Building Bridges II. Bolyai Society Mathematical Studies, vol 28. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59204-5_6

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