## Abstract

We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let G ~ G(n, 1/2, k) be a random graph on n vertices with a planted clique of size k. We show that no algorithm that makes at most q = o(n^{2}/k^{2} + n) adaptive queries to the adjacency matrix of G is likely to find the planted clique. On the other hand, when k ≥ (2 + ε) log_{2} n there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making q = O(n^{2}/k^{2}) log^{2} n + n log n) adaptive queries. For detection, the additive n term is not necessary: the number of queries needed to detect the presence of a planted clique is n^{2}/k^{2} (up to logarithmic factors).

Original language | English (US) |
---|---|

Pages (from-to) | 759-773 |

Number of pages | 15 |

Journal | Alea |

Volume | 17 |

Issue number | 2 |

DOIs | |

State | Published - 2020 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability

## Keywords

- Adaptive query model
- Planted clique
- Random graph