Computer Science/Discrete Mathematics Seminar I | |

Topic: | Association Schemes, Non-Commutative Polynomials and Lasserre Lower Bounds for Planted Clique |

Speaker: | Raghu Meka |

Affiliation: | DIMACS (Rutgers); Member, School of Mathematics |

Date: | Monday, May 13 |

Time/Room: | 1:30pm - 3:30pm/S-101 |

Video Link: | https://video.ias.edu/csdm/1213/0513-RaghuMeka |

Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size k is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for k ~ sqrt(n). Here we show that beating sqrt(n) would require substantially new algorithmic ideas, by proving a lower bound for the problem in the Lasserre hierarchy, the most powerful class of semi-de finite programming algorithms we know of. Our (average case) lower bound uses tools from the classical theory of association schemes and some new large deviation bounds for matrix-valued polynomials which could be of independent interest. Joint work with Avi Wigderson