April 2023 Sequential importance sampling for estimating expectations over the space of perfect matchings
Yeganeh Alimohammadi, Persi Diaconis, Mohammad Roghani, Amin Saberi
Author Affiliations +
Ann. Appl. Probab. 33(2): 999-1033 (April 2023). DOI: 10.1214/22-AAP1834


This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a (1±ϵ)-approximation for the number of perfect matchings of a λ-dense bipartite graph, using O(n12λλϵ2) samples. With size n on each side and for 12>λ>0, a λ-dense bipartite graph has all degrees greater than (λ+12)n.

Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements.

Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.

Funding Statement

Yeganeh Alimohnammadi, Mohammad Roghani, and Amin Saberi are supported by NSF grant CCF1812919.


The authors thank Sourav Chatterjee, for helpful feedback on the early version of the manuscript and the proof of Lemma 2.2, and Jan Vondrák for discussions on concentration inequalities. Also, we would like to thank Nick Wormald and Fredrick Manners for bringing us up to speed about Latin squares. We thank our anonymous reviewers for their insightful comments and suggestions.


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Yeganeh Alimohammadi. Persi Diaconis. Mohammad Roghani. Amin Saberi. "Sequential importance sampling for estimating expectations over the space of perfect matchings." Ann. Appl. Probab. 33 (2) 999 - 1033, April 2023. https://doi.org/10.1214/22-AAP1834


Received: 1 July 2021; Revised: 1 April 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 1515.60035
MathSciNet: MR4564419
Digital Object Identifier: 10.1214/22-AAP1834

Primary: 65C05
Secondary: 05B15

Keywords: card guessing experiment , dense bipartite graph , KL-divergence , Latin squares , perfect matching , sequential importance sampling , Stochastic block model

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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