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
MetadataShow full item record
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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
Institute of Electrical and Electronics Engineers (IEEE)
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