ECCC-Report TR16-058https://eccc.weizmann.ac.il/report/2016/058Comments and Revisions published for TR16-058en-usTue, 12 Apr 2016 20:53:49 +0300
Paper TR16-058
| A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem |
Boaz Barak,
Samuel Hopkins,
Jonathan Kelner,
Pravesh Kothari,
Ankur Moitra,
Aaron Potechin
https://eccc.weizmann.ac.il/report/2016/058We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\log n)^{1/2}}$ for some constant $c>0$.
This yields a nearly tight $n^{1/2 - o(1)}$ bound on the value of this program for any degree $d = o(\log n)$. Moreover we introduce a new framework that we call \emph{pseudo-calibration} to construct Sum of Squares lower bounds.
This framework is inspired by taking a computational analog 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. Tue, 12 Apr 2016 20:53:49 +0300https://eccc.weizmann.ac.il/report/2016/058