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Counting Perfect Matchings and Hamiltonian Cycles Faster

Author Baitian Li

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LIPIcs.ICALP.2026.138.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • permanent
  • hafnian
  • Hamiltonian cycle
  • Kakeya sets

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Abstract

We show that the hafnian of a symmetric 2n× 2n matrix of poly(n)-bit integers (which counts the number of perfect matchings of a 2n-vertex graph) and the number of Hamiltonian cycles of an n-vertex directed graph can be computed in time 2^{n-Ω(√n)}, improving and generalizing an earlier algorithm of Björklund, Kaski, and Williams (Algorithmica 2019) that runs in time 2^{n - Ω(√{n/log log n})}.
A key tool of our approach is the design of a data structure that supports fast evaluation of high-order derivatives of hafnian and Hamiltonian cycles, which integrates with the new approach on multivariate multipoint evaluation by Bhargava, Ghosh, Guo, Kumar, and Umans (FOCS 2022, JACM 2024).

Cite As Get BibTex

Baitian Li. Counting Perfect Matchings and Hamiltonian Cycles Faster. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 138:1-138:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.138

Author Details

Baitian Li
  • Columbia University, New York, NY, USA

Funding

  • Li, Baitian: Supported in part by NSF Grant CCF-2238221, a Packard Foundation Fellowship, and a Columbia SEAS Presidential Fellowship.

Acknowledgements

The work was done when the author was an undergraduate student at Tsinghua University. The author would like to thank Josh Alman and anonymous referees for helpful comments on earlier drafts.

References

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