Finding tight Hamilton cycles in random hypergraphs faster

TL;DR

This paper introduces a deterministic polynomial-time algorithm that efficiently finds tight Hamilton cycles in random hypergraphs at lower edge probabilities than previous methods, advancing the understanding of hypergraph Hamiltonicity.

Contribution

It provides the first deterministic polynomial-time algorithm for finding tight Hamilton cycles in random hypergraphs at edge probability $C rac{ ext{polylog}(n)}{n}$, improving upon prior randomized algorithms.

Findings

Algorithm works for $p o C rac{ ext{polylog}(n)}{n}$

Deterministic polynomial-time complexity

Partially answers Dudek and Frieze's question

Abstract

In an $r$-uniform hypergraph on $n$ vertices a tight Hamilton cycle consists of $n$ edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of $r$ vertices. We provide a first deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton cycles in random $r$-uniform hypergraphs with edge probability at least $C \log^3n/n$. Our result partially answers a question of Dudek and Frieze [Random Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton cycles exists already for $p=\omega(1/n)$ for $r=3$ and $p=(e + o(1))/n$ for $r\ge 4$ using a second moment argument. Moreover our algorithm is superior to previous results of Allen, B\"ottcher, Kohayakawa and Person [Random Structures & Algorithms 46 (2015), 446-465] and Nenadov and \v{S}kori\'c [arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a randomised polynomial time algorithm working for edge probabilities $p\ge n^{-1+\varepsilon}$, while the algorithm of Nenadov and \v{S}kori\'c is a randomised quasipolynomial time algorithm working for edge probabilities $p\ge C\log^8n/n$.