An Asymptotically Fast Polynomial Space Algorithm for Hamiltonicity Detection in Sparse Directed Graphs

TL;DR

This paper introduces a polynomial space Monte Carlo algorithm that detects Hamiltonian cycles in sparse directed graphs more efficiently, achieving a time complexity that asymptotically matches the best exponential space algorithms.

Contribution

It combines inclusion-exclusion with linear algebra techniques to improve polynomial space algorithms for Hamiltonicity detection in sparse directed graphs.

Findings

Achieves $2^{n-rac{n}{ ext{delta}}}$ time complexity for sparse graphs.

Matches the fastest known exponential space algorithms asymptotically.

Improves upon previous polynomial space algorithms in terms of efficiency.

Abstract

We present a polynomial space Monte Carlo algorithm that given a directed graph on $n$ vertices and average outdegree $\delta$, detects if the graph has a Hamiltonian cycle in $2^{n-\Omega(\frac{n}{\delta})}$ time. This asymptotic scaling of the savings in the running time matches the fastest known exponential space algorithm by Bj\"orklund and Williams ICALP 2019. By comparison, the previously best polynomial space algorithm by Kowalik and Majewski IPEC 2020 guarantees a $2^{n-\Omega(\frac{n}{2^\delta})}$ time bound. Our algorithm combines for the first time the idea of obtaining a fingerprint of the presence of a Hamiltonian cycle through an inclusion--exclusion summation over the Laplacian of the graph from Bj\"orklund, Kaski, and Koutis ICALP 2017, with the idea of sieving for the non-zero terms in an inclusion--exclusion summation by listing solutions to systems of linear equations over $\mathbb{Z}_2$ from Bj\"orklund and Husfeldt FOCS 2013.