## Abstract

We design new polynomial-time algorithms for recovering planted cliques in the semi-random graph model introduced by Feige and Kilian. The previous best algorithms for this model succeed if the planted clique has size at least n2/3 in a graph with n vertices. Our algorithms work for planted-clique sizes approaching n1/2-the information-theoretic threshold in the semi-random model and a conjectured computational threshold even in the easier fully-random model. This result comes close to resolving open questions by Feige and Steinhardt. To generate a graph in the semi-random planted-clique model, we first 1) plant a clique of size k in an n-vertex-graph with edge probability 1/2 and then adversarially add or delete an arbitrary number edges not touching the planted clique and delete any subset of edges going out of the planted clique. For every ">0, we give an nO(1/")-time algorithm that recovers a clique of size k in this model whenever k ≥ n1/2+". In fact, our algorithm computes, with high probability, a list of about n/k cliques of size k that contains the planted clique. Our algorithms also extend to arbitrary edge probabilities p and improve on the previous best guarantee whenever p ≤ 1-n-0.001. Our algorithms rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite-random graphs into algorithms for semi-random planted clique. Analogous to the (conjecturally) optimal algorithms for the fully-random model, the previous best guarantees for semi-random planted clique correspond to spectral relaxations of biclique numbers based on eigenvalues of adjacency matrices. We construct an SDP lower bound that shows that the n2/3 threshold in prior works is an inherent limitation of these spectral relaxations. We go beyond this limitation by using higher-order sum-of-squares relaxations for biclique numbers. We also provide some evidence that the information-computation trade-off of our current algorithms may be inherent by proving an average-case lower bound for unbalanced bicliques in the low-degree polynomial model.

Original language | English (US) |
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Title of host publication | STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing |

Editors | Barna Saha, Rocco A. Servedio |

Publisher | Association for Computing Machinery |

Pages | 1918-1926 |

Number of pages | 9 |

ISBN (Electronic) | 9781450399135 |

DOIs | |

State | Published - Jun 2 2023 |

Externally published | Yes |

Event | 55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States Duration: Jun 20 2023 → Jun 23 2023 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 55th Annual ACM Symposium on Theory of Computing, STOC 2023 |
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Country/Territory | United States |

City | Orlando |

Period | 6/20/23 → 6/23/23 |

## All Science Journal Classification (ASJC) codes

- Software

## Keywords

- planted clique
- semi-random
- semidefinite programming
- sum-of-squares hierarchy