Finding Planted Cycles in a Random Graph

TL;DR

This paper investigates the detectability of planted cycles in random graphs, establishing precise thresholds for information-theoretic possibility and providing a polynomial-time algorithm for almost exact recovery, highlighting a contrast with the planted clique problem.

Contribution

It introduces a threshold for almost-exact recovery of planted cycles in Erdős-Rényi graphs and presents a polynomial-time algorithm achieving this, contrasting with known hardness results for similar problems.

Findings

Recovery is possible below a certain edge density threshold.

A polynomial-time algorithm achieves almost exact recovery.

There is a sharp phase transition in recoverability based on parameters.

Abstract

In this paper, we study the problem of finding a collection of planted cycles in an \ER random graph $G \sim \mathcal{G}(n, \lambda/n)$, in analogy to the famous Planted Clique Problem. When the cycles are planted on a uniformly random subset of $\delta n$ vertices, we show that almost-exact recovery (that is, recovering all but a vanishing fraction of planted-cycle edges as $n \to \infty$) is information-theoretically possible if $\lambda < \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$ and impossible if $\lambda > \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$. Moreover, despite the worst-case computational hardness of finding long cycles, we design a polynomial-time algorithm that attains almost exact recovery when $\lambda < \frac{1}{(\sqrt{2 \delta} + \sqrt{1-\delta})^2}$. This stands in stark contrast to the Planted Clique Problem, where a significant computational-statistical gap is widely conjectured.