Finding Hamilton cycles in random graphs with few queries

TL;DR

This paper studies the minimal number of adjacency queries needed to find Hamilton cycles in random graphs, showing that near-linear queries suffice in a certain probability regime, with results tight in both parameters.

Contribution

It introduces a new query-based framework for finding Hamilton cycles in random graphs and establishes tight bounds on the number of queries required.

Findings

Hamilton cycles can be found with high probability after exposing about n edges

The number of queries needed is tight in the probability regime considered

The results establish bounds that are optimal in both edge exposure and probability parameters

Abstract

We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of ${\mathcal G}(n,p)$ in order to typically find a subgraph possessing a given target property. We show that if $p\geq \frac{\ln n+\ln\ln n+\omega(1)}{n}$, then one can find a Hamilton cycle with high probability after exposing $(1+o(1))n$ edges. Our result is tight in both $p$ and the number of exposed edges.