2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) (2016)
New Brunswick, New Jersey, USA
Oct. 9, 2016 to Oct. 11, 2016
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2016.53
We prove that with high probability over the choice of a random graph G from the Erdős-Rényi distribution G(n,1/2), the no(d)-time degree d Sum-of-Squares 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 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.
Bayes methods, Observers, Programming, Complexity theory, Algorithm design and analysis
B. Barak, S. B. Hopkins, J. Kelner, P. Kothari, A. Moitra and A. Potechin, "A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem," 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), New Brunswick, New Jersey, USA, 2016, pp. 428-437.