Finding Planted Cycles in a Random Graph

TL;DR

This work studies the planted 2-factor problem in ER graphs G$(n,\lambda/n)$ with a random planted vertex subset of size $\delta n$. It establishes a sharp phase transition for almost exact recovery at the threshold $\lambda<\frac{1}{(\sqrt{2\delta}+\sqrt{1-\delta})^2}$ and proves impossibility above it. The authors introduce a generating function framework and a greedy trail-based algorithm that achieve almost exact recovery in the feasible regime, showing no computational-statistical gap relative to planted clique heuristics. The analysis combines generating-function counts of $(a,b)$-trails, branching-process techniques, and a sprinkling argument to connect trees into long balanced cycles, offering a detailed All-regime picture and highlighting open questions about intermediate regimes and generalizations.

Abstract

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

Figures (4)

• Figure 1: An example where $H^*$ is a $4$-cycle with red solid edges, $H$ is a $4$-cycle with blue dashed edges, and $H^*\Delta H$ has a $(3,3)$-Eulerian circuit.
• Figure 2: A schematic representation of a path tree with a depth of $2$. The red nodes are called "hub" nodes, and the dashed purple lines represent $(m^{\ast}, m^{\ast})$-paths, where $m^{\ast}$ is given in Lemma \ref{['lmm:m-star']}. A purple path along with the hub node below it is referred to as a "path layer" of the path tree.
• Figure 3: The solid purple line represents a path of length strictly less than $2m^{\ast}$.
• Figure 4: Five edge construction (reproduced from gaudio2025all).

Theorems & Definitions (56)

• Definition 1.1: Planted cycles model
• Theorem 1.1
• Remark 1.1
• Lemma 2.1
• proof : Proof of Lemma \ref{['lemma:difference-graph']}
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof