## A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

##### Author(s)

Barak, Boaz; Hopkins, Samuel B.; Kothari, Pravesh; Potechin, Aaron; Moitra, Ankur; Kelner, Jonathan Adam; ... Show more Show less
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Show full item record##### Abstract

We prove that with high probability over the choice of a random graph G from the Erds-Rényi distribution G(n,1/2), the n[superscript o(d)]-time degree d Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least n[superscript 1/2-c(d/log n)1/2] for some constant c > 0. This yields a nearly tight n[superscript 1/2-o(1))] bound on the value of this program for any degree d = o(log n). Moreover we introduce a new framework that we call pseudo-calibration to construct Sum-of-Squares lower bounds. This framework is inspired by taking a computational analogue of Bayesian probability theory. It yields a general recipe for constructing good pseudo-distributions (i.e., dual certificates for the Sum-of-Squares semidefinite program), and sheds further light on the ways in which this hierarchy differs from others.

##### Date issued

2016-12##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

##### Publisher

Institute of Electrical and Electronics Engineers (IEEE)

##### Citation

Barak, Boaz, et al. "A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem." 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), 9-11 October, 2016, New Brunswick, New Jersey, IEEE, 2016 © 2016 IEEE

Version: Original manuscript

##### ISBN

978-1-5090-3933-3