Counting perfect matchings and Hamiltonian cycles faster

TL;DR

This paper presents an improved algorithm for counting perfect matchings and Hamiltonian cycles in graphs, achieving faster computation times using advanced data structures and multivariate evaluation techniques.

Contribution

The authors develop a new algorithm that improves the time complexity for counting perfect matchings and Hamiltonian cycles, generalizing previous methods.

Findings

Counting perfect matchings can be done in time 2^{n - Ω(√n)}.

Counting Hamiltonian cycles can be done in time 2^{n - Ω(√n)}.

The approach uses a novel data structure for high-order derivatives evaluation.

Abstract

We show that the hafnian of a symmetric $2n\times 2n$ matrix of $\operatorname{poly}(n)$-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 $2^{n-\Omega(\sqrt{n})}$, improving and generalizing an earlier algorithm of Bj\"orklund, Kaski, and Williams (Algorithmica 2019) that runs in time $2^{n - \Omega\left(\sqrt{n/\log \log n}\right)}$. 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).