Finding paths in sparse random graphs requires many queries

TL;DR

This paper establishes lower bounds on the number of adjacency queries needed by algorithms to find long paths in sparse random graphs, showing that significantly more queries than the path length are necessary.

Contribution

It proves tight lower bounds on query complexity for finding long paths in sparse Erdős–Rényi graphs, highlighting fundamental limitations of query-based algorithms.

Findings

Any algorithm must query at least Ω(ℓ/(pε log(1/ε))) pairs to find a path of length ℓ.

Finding long paths requires significantly more queries than the path length.

The bounds are tight up to a logarithmic factor.

Abstract

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph $G\sim {\mathcal G}(n,p)$ in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in $G\sim \mathcal G(n,p)$ when $p=\frac{1+\varepsilon}{n}$ for some fixed constant $\varepsilon>0$. This random graph is known to have typically linearly long paths. To have $\ell$ edges with high probability in $G\sim \mathcal G(n,p)$ one clearly needs to query at least $\Omega\left(\frac{\ell}{p}\right)$ pairs of vertices. Can we find a path of length $\ell$ economically, i.e., by querying roughly that many pairs? We argue that this is not possible and one needs to query significantly more pairs. We prove that any randomised algorithm which finds a path of length $\ell=\Omega\left(\frac{\log\left(\frac{1}{\varepsilon}\right)}{\varepsilon}\right)$ with at least constant probability in $G\sim \mathcal G(n,p)$ with $p=\frac{1+\varepsilon}{n}$ must query at least $\Omega\left(\frac{\ell}{p\varepsilon \log\left(\frac{1}{\varepsilon}\right)}\right)$ pairs of vertices. This is tight up to the $\log\left(\frac{1}{\varepsilon}\right)$ factor.