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On Finding Randomly Planted Cliques in Arbitrary Graphs

Authors Francesco Agrimonti , Marco Bressan , Tommaso d'Orsi

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Mathematics of computing → Discrete mathematics
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Computational Complexity
  • Planted Clique
  • Semi-random
  • Unique Games Conjecture
  • Approximation Algorithms

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Abstract

We study a planted clique model introduced by Feige [Uriel Feige, 2021] where a complete graph of size c⋅ 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) ⋅ 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,1/2)+K_√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.

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Francesco Agrimonti, Marco Bressan, and Tommaso d'Orsi. On Finding Randomly Planted Cliques in Arbitrary Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.11

Author Details

Francesco Agrimonti
  • Parma, Italy
Marco Bressan
  • Department of Computer Science, University of Milan, Italy
Tommaso d'Orsi
  • Department of Computing Sciences, Bocconi University, Milan, Italy

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