Improved Approximation for the Number of Hamiltonian Cycles in Dense Digraphs

TL;DR

This paper introduces an improved algorithm for approximating the number of Hamiltonian cycles in dense directed graphs, achieving significantly faster expected running time than previous methods by leveraging sequential acceptance/rejection techniques.

Contribution

It presents a novel approximation algorithm based on sequential acceptance/rejection, providing a new bound on Hamiltonian cycles and outperforming existing Markov chain methods in dense graphs.

Findings

Expected running time of O(n^{8.5}) for dense graphs

Improves previous methods by at least n^{4.5}( ext{log} n)^4 in efficiency

Establishes that counting Hamiltonian cycles in dense graphs is #P-Complete

Abstract

We propose an improved algorithm for counting the number of Hamiltonian cycles in a directed graph. The basic idea of the method is sequential acceptance/rejection, which is successfully used in approximating the number of perfect matchings in dense bipartite graphs. As a consequence, a new bound on the number of Hamiltonian cycles in a directed graph is proved, by using the ratio of the number of 1-factors. Based on this bound, we prove that our algorithm runs in expected time of $O(n^{8.5})$ for dense problems. This improves the Markov chain method, the most powerful existing method, a factor of at least $n^{4.5}(\log n)^{4}$ in running time. This class of dense problems is shown to be nontrivial in counting, in the sense that it is $#$P-Complete.