Finding Planted Cycles in a Random Graph
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908632883396608 |
|---|---|
| author | Gaudio, Julia Sandon, Colin Xu, Jiaming Yang, Dana |
| author_facet | Gaudio, Julia Sandon, Colin Xu, Jiaming Yang, Dana |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finding Planted Cycles in a Random Graph Gaudio, Julia Sandon, Colin Xu, Jiaming Yang, Dana Statistics Theory Probability 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. |
| title | Finding Planted Cycles in a Random Graph |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2511.04058 |