A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem
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17m1138236.pdf
Description
Published version
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904.49 KB
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Adobe PDF
Checksum (MD5)
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Author(s)
Barak, Boaz
Hopkins, Samuel
Kelner, Jonathan
Kothari, Pravesh K
Moitra, Ankur
Potechin, Aaron
Date Issued
2019
Journal
SIAM Journal on Computing
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Barak, Boaz, Hopkins, Samuel, Kelner, Jonathan, Kothari, Pravesh K, Moitra, Ankur et al. 2019. "A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem." SIAM Journal on Computing, 48 (2).
Version
Final published version
Abstract
© 2019 Society for Industrial and Applied Mathematics We prove that with high probability over the choice of a random graph G from the Erd\H os-Rényi distribution G(n, 1/2), the nO(d)-time degree d sum-of-squares (SOS) semidefinite programming relaxation for the clique problem will give a value of at least n1/2 - c(d/ log n)1/2 for some constant c > 0. This yields a nearly tight n1/2 - 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 pseudocalibration to construct SOS lower bounds. This framework is inspired by taking a computational analogue of Bayesian probability theory. It yields a general recipe for constructing good pseudodistributions (i.e., dual certificates for the SOS 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
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DOI of Published Version
https://doi.org/10.1137/17M1138236
