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On Finding Randomly Planted Cliques in Arbitrary Graphs

Francesco Agrimonti, Marco Bressan, Tommaso d'Orsi

TL;DR

This work analyzes random planting of a clique inside an arbitrary graph and introduces a simple deterministic algorithm that recovers a substantial planted clique under a maximum-degree constraint, running in near-linear time. The key idea is to exploit the correlation between vertex degrees (via slackness) and the planted clique, formalized through a slackness profile and a densification lemma that creates a dense, planted-related region to extract a clique. The results extend beyond cliques to balanced bicliques, yielding significantly larger structures than worst-case guarantees. A lower-bound construction shows limits of degree-based densification when the planted fraction is small (c < 1/2), illustrating the boundary between this average/w semi-random model and worst-case hardness, as well as the potential algorithmic separation under UGC. Overall, the paper demonstrates that randomness in the planting location can enable efficient recovery in a broad class of graphs, highlighting a gap between typical instances and worst-case instances.

Abstract

We study a planted clique model introduced by Feige where a complete graph of size $c\cdot n$ is planted uniformly at random in an arbitrary $n$-vertex graph. We give a simple deterministic algorithm that, in almost linear time, recovers a clique of size $(c/3)^{O(1/c)} \cdot n$ as long as the original graph has maximum degree at most $(1-p)n$ for some fixed $p>0$. The proof hinges on showing that the degrees of the final graph are correlated with the planted clique, in a way similar to (but more intricate than) the classical $G(n,\frac{1}{2})+K_{\sqrt{n}}$ planted clique model. Our algorithm suggests a separation from the worst-case model, where, assuming the Unique Games Conjecture, no polynomial algorithm can find cliques of size $Ω(n)$ for every fixed $c>0$, even if the input graph has maximum degree $(1-p)n$. Our techniques extend beyond the planted clique model. For example, when the planted graph is a balanced biclique, we recover a balanced biclique of size larger than the best guarantees known for the worst case.

On Finding Randomly Planted Cliques in Arbitrary Graphs

TL;DR

This work analyzes random planting of a clique inside an arbitrary graph and introduces a simple deterministic algorithm that recovers a substantial planted clique under a maximum-degree constraint, running in near-linear time. The key idea is to exploit the correlation between vertex degrees (via slackness) and the planted clique, formalized through a slackness profile and a densification lemma that creates a dense, planted-related region to extract a clique. The results extend beyond cliques to balanced bicliques, yielding significantly larger structures than worst-case guarantees. A lower-bound construction shows limits of degree-based densification when the planted fraction is small (c < 1/2), illustrating the boundary between this average/w semi-random model and worst-case hardness, as well as the potential algorithmic separation under UGC. Overall, the paper demonstrates that randomness in the planting location can enable efficient recovery in a broad class of graphs, highlighting a gap between typical instances and worst-case instances.

Abstract

We study a planted clique model introduced by Feige where a complete graph of size is planted uniformly at random in an arbitrary -vertex graph. We give a simple deterministic algorithm that, in almost linear time, recovers a clique of size as long as the original graph has maximum degree at most for some fixed . The proof hinges on showing that the degrees of the final graph are correlated with the planted clique, in a way similar to (but more intricate than) the classical planted clique model. Our algorithm suggests a separation from the worst-case model, where, assuming the Unique Games Conjecture, no polynomial algorithm can find cliques of size for every fixed , even if the input graph has maximum degree . Our techniques extend beyond the planted clique model. For example, when the planted graph is a balanced biclique, we recover a balanced biclique of size larger than the best guarantees known for the worst case.
Paper Structure (9 sections, 13 theorems, 42 equations, 1 table, 3 algorithms)

This paper contains 9 sections, 13 theorems, 42 equations, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Results
  3. Techniques
  4. Preliminaries
  5. Slackness profile and densification
  6. Application to clique
  7. Application to balanced biclique
  8. A lower bound on densification
  9. Concentration inequalities

Key Result

Theorem 1.2

There exists a deterministic algorithm $\mathcal{A}$ with the following guarantees. For every $c \in (0,1)$ and every $n$-vertex graph $G$, if $\hat{G}\sim \mathcal{G}\mathopen{}\mathclose{\left(G,K_{cn}\right)$ then $\mathcal{A}(\hat{G})$ with probability at least $1-\tfrac{1}}{n^2}$ returns a cliq where $p = 1- \frac{\Delta}{n}$ and $\Delta$ is the maximum degree of $G$. Moreover $\mathcal{A}$ r

Theorems & Definitions (25)

  • Definition 1.1: Random planting in arbitrary graphs
  • Theorem 1.2: Simplified version
  • Theorem 1.3: Simplified version
  • Remark 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Definition 4.1: Bulging set
  • Lemma 4.2
  • proof
  • Lemma 4.3: Densification Lemma
  • ...and 15 more