The Parity of Directed Hamiltonian Cycles

TL;DR

This paper introduces efficient algorithms to determine the parity of Hamiltonian cycles in directed graphs and bipartite graphs, utilizing a novel combinatorial formula to improve computational complexity.

Contribution

It presents the first deterministic polynomial-space algorithm for parity of Hamiltonian cycles in directed graphs and a faster expected-time algorithm for bipartite graphs based on a new combinatorial formula.

Findings

Deterministic algorithm computes parity in O(1.619^n) time

Expected-time algorithm for bipartite graphs runs in 1.5^n poly(n)

New combinatorial formula for Hamiltonian cycles modulo an integer

Abstract

We present a deterministic algorithm that given any directed graph on n vertices computes the parity of its number of Hamiltonian cycles in O(1.619^n) time and polynomial space. For bipartite graphs, we give a 1.5^n poly(n) expected time algorithm. Our algorithms are based on a new combinatorial formula for the number of Hamiltonian cycles modulo a positive integer.