A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem
Name
1604.03084v2.pdf
Size
708.24 KB
Format
Adobe PDF
Checksum (MD5)
4b19cd0eb9c639e37d5c1adf09771de8
Author(s)
Barak, Boaz
Hopkins, Samuel B.
Kothari, Pravesh
Potechin, Aaron
Moitra, Ankur
Kelner, Jonathan Adam
Date Issued
December 2016
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
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1109/FOCS.2016.53
