### Abstract

The problem of finding large cliques in random graphs and its “planted” variant, where one wants to recover a clique of size ω log (n) added to an Erdős-Rényi graph G ∼ G(n,^{1} _{2} ), have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size ω = Ω(n). By contrast, information theoretically, one can recover planted cliques so long as ω log (n). In this work, we continue the investigation of algorithms from the Sum of Squares hierarchy for solving the planted clique problem begun by Meka, Potechin, and Wigderson [2] and Deshpande and Montanari [25]. Our main result is that degree four SoS does not recover the planted clique unless ω n/ polylog n, improving on the bound ω n^{1}/^{3} due to Reference [25]. An argument of Kelner shows that the this result cannot be proved using the same certificate as prior works. Rather, our proof involves constructing and analyzing a new certificate that yields the nearly tight lower bound by “correcting” the certificate of References [2, 25, 27].

Original language | English (US) |
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Article number | 28 |

Journal | ACM Transactions on Algorithms |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - Jul 2018 |

### All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)

### Keywords

- Planted clique
- Random matrices
- Sum of Squares method

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## Cite this

*ACM Transactions on Algorithms*,

*14*(3), [28]. https://doi.org/10.1145/3178538