Sum-of-squares Lower Bounds for Planted Clique

@article{Meka2013SumofsquaresLB,
  title={Sum-of-squares Lower Bounds for Planted Clique},
  author={Raghu Meka and Aaron Potechin and A. Wigderson},
  journal={Proceedings of the forty-seventh annual ACM symposium on Theory of Computing},
  year={2013}
}
Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size k is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for k = Θ(√n). In this paper we study the complexity of the planted clique problem under algorithms from the Sum-Of-Squares hierarchy. We prove the first average case lower bound for this model: for… Expand
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References

SHOWING 1-10 OF 81 REFERENCES
Finding a large hidden clique in a random graph
We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n, 1/2), and then choose randomly a subset Q of vertices of size k and force it to be aExpand
Finding Hidden Cliques in Linear Time with High Probability
TLDR
A new algorithm for finding hidden cliques that both runs in time O(n2) (that is, linear in the size of the input) and has a failure probability that tends to 0 as n tends to ∞ is presented. Expand
Expected Complexity of Graph Partitioning Problems
  • L. Kucera
  • Computer Science, Mathematics
  • Discret. Appl. Math.
  • 1995
TLDR
The expected time complexity of two graph partitioning problems: the graph coloring and the cut into equal parts is studied to obtain a sublinear expected time algorithm for k-coloring of k-colorable graphs. Expand
When are small subgraphs of a random graph normally distributed
A random graph K(n, p) is a graph on the vertex set {1 . . . . . n} whose edges appear independently from each other and with probability p=p(n). One of the classical questions qf the theory ofExpand
Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph
TLDR
The results indicate that approximating Densest k-subgraph within a polynomial factor might be a harder problem than Unique Games or Small Set Expansion, since these problems were recently shown to be solvable using neΩ(1) rounds of the Lasserre hierarchy, where e is the completeness parameter in Unique Games and Small Set expansion. Expand
Large Cliques Elude the Metropolis Process
  • M. Jerrum
  • Mathematics, Computer Science
  • Random Struct. Algorithms
  • 1992
TLDR
It is shown that the Metropolis process takes super-polynomial time to locate a clique that is only slightly bigger than that produced by the greedy heuristic, which is one step above the greedy one in its level of sophistication. Expand
Hypercontractivity, sum-of-squares proofs, and their applications
TLDR
Reductions between computing the 2->4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory and the study of Khot's Unique Games Conjecture are shown. Expand
Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Graph Partitioning and Quadratic Integer Programming with PSD Objectives
  • V. Guruswami, A. Sinop
  • Mathematics, Computer Science
  • 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
  • 2011
TLDR
An approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semi definite objective functions and global linear constraints is presented, and an algorithm for independent sets in graphs that performs well when the Laplacian does not have too many eigenvalues bigger than $1+o(1). Expand
The Probable Value of the Lovász--Schrijver Relaxations for Maximum Independent Set
TLDR
It follows that for those relaxations known to be efficiently computable, namely, for r=O(1), the value of the relaxation is comparable to the theta function. Expand
A Large Deviation Result on the Number of Small Subgraphs of a Random Graph
  • V. Vu
  • Computer Science, Mathematics
  • Combinatorics, Probability and Computing
  • 2001
Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions.Expand
...
1
2
3
4
5
...