A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

We prove that with high probability over the choice of a random graph G from the Erdös-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.

[1]  Andrew Granville,et al.  HARALD CRAM ER AND THE DISTRIBUTION OF PRIME NUMBERS , 1993 .

[2]  Terence Tao,et al.  The dichotomy between structure and randomness, arithmetic progressions, and the primes , 2005, math/0512114.

[3]  Pravesh Kothari,et al.  SoS and Planted Clique: Tight Analysis of MPW Moments at all Degrees and an Optimal Lower Bound at Degree Four , 2015, ArXiv.

[4]  Andrea Montanari,et al.  Improved Sum-of-Squares Lower Bounds for Hidden Clique and Hidden Submatrix Problems , 2015, COLT.

[5]  Jean B. Lasserre,et al.  An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs , 2001, IPCO.

[6]  Ludek Kucera,et al.  Expected Complexity of Graph Partitioning Problems , 1995, Discret. Appl. Math..

[7]  Dima Grigoriev,et al.  Complexity of Positivstellensatz proofs for the knapsack , 2002, computational complexity.

[8]  Bruce E. Hajek,et al.  Computational Lower Bounds for Community Detection on Random Graphs , 2014, COLT.

[9]  Joan Feigenbaum,et al.  On the random-self-reducibility of complete sets , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[10]  Avi Wigderson,et al.  Sum-of-Squares Lower Bounds for Sparse PCA , 2015, NIPS.

[11]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[12]  David Steurer,et al.  Rounding sum-of-squares relaxations , 2013, Electron. Colloquium Comput. Complex..

[13]  Mark Braverman,et al.  Inapproximability of NP-Complete Variants of Nash Equilibrium , 2011, Theory Comput..

[14]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[15]  Aaron Potechin,et al.  Bounds on the Norms of Uniform Low Degree Graph Matrices , 2016, APPROX-RANDOM.

[16]  David Steurer,et al.  Sum-of-squares proofs and the quest toward optimal algorithms , 2014, Electron. Colloquium Comput. Complex..

[17]  Prasad Raghavendra,et al.  On the Integrality Gap of Degree-4 Sum of Squares for Planted Clique , 2016, SODA.

[18]  Ari Juels,et al.  Hiding Cliques for Cryptographic Security , 1998, SODA '98.

[19]  Pascal Koiran,et al.  Hidden Cliques and the Certification of the Restricted Isometry Property , 2012, IEEE Transactions on Information Theory.

[20]  E. Szemerédi.  Regular Partitions of Graphs , 1975 .

[21]  Noga Alon,et al.  Finding a large hidden clique in a random graph , 1998, SODA '98.

[22]  N. Z. Shor.  Class of global minimum bounds of polynomial functions , 1987 .

[23]  Avi Wigderson,et al.  Public-key cryptography from different assumptions , 2010, STOC '10.

[24]  利久 亀井,et al.  California Institute of Technology , 1958, Nature.

[25]  Robert Krauthgamer,et al.  How hard is it to approximate the best Nash equilibrium? , 2009, SODA.

[26]  Philippe Rigollet,et al.  Complexity Theoretic Lower Bounds for Sparse Principal Component Detection , 2013, COLT.

[27]  Avi Wigderson,et al.  Sum-of-squares Lower Bounds for Planted Clique , 2015, STOC.

[28]  Yuan Zhou,et al.  Hypercontractivity, sum-of-squares proofs, and their applications , 2012, STOC '12.

[29]  Alan M. Frieze,et al.  The regularity lemma and approximation schemes for dense problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[30]  S. Shen-Orr,et al.  Network motifs: simple building blocks of complex networks. , 2002, Science.

[31]  Noga Alon,et al.  Testing k-wise and almost k-wise independence , 2007, STOC '07.

[32]  Pravesh Kothari,et al.  Sum of Squares Lower Bounds from Pairwise Independence , 2015, STOC.

[33]  Robert Krauthgamer,et al.  The Probable Value of the Lovász--Schrijver Relaxations for Maximum Independent Set , 2003, SIAM J. Comput..

[34]  Luca Trevisan,et al.  On Worst-Case to Average-Case Reductions for NP Problems , 2005, Electron. Colloquium Comput. Complex..

[35]  Mark Jerrum,et al.  Large Cliques Elude the Metropolis Process , 1992, Random Struct. Algorithms.

[36]  David Steurer,et al.  Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method , 2014, STOC.

[37]  Grant Schoenebeck,et al.  Linear Level Lasserre Lower Bounds for Certain k-CSPs , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.