A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
A PCP characterization of NP with optimal amortized query complexity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Finding Dense Subgraphs with Semidefinite Programming
APPROX '98 Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization
Improved Inapproximability Results for MaxClique, Chromatic Number and Approximate Graph Coloring
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
On the densest k-subgraph problems
On the densest k-subgraph problems
A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming
Mathematics of Operations Research
Ruling Out PTAS for Graph Min-Bisection, Densest Subgraph and Bipartite Clique
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Local versus global properties of metric spaces
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut
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A Linear Round Lower Bound for Lovasz-Schrijver SDP Relaxations of Vertex Cover
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Linear programming relaxations of maxcut
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation Algorithms Using Hierarchies of Semidefinite Programming Relaxations
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Integrality gaps of 2 - o(1) for Vertex Cover SDPs in the Lovész-Schrijver Hierarchy
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Local Global Tradeoffs in Metric Embeddings
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Improved Approximation Guarantees through Higher Levels of SDP Hierarchies
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Linear Level Lasserre Lower Bounds for Certain k-CSPs
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Integrality gaps for Sherali-Adams relaxations
Proceedings of the forty-first annual ACM symposium on Theory of computing
CSP gaps and reductions in the lasserre hierarchy
Proceedings of the forty-first annual ACM symposium on Theory of computing
Optimal Sherali-Adams Gaps from Pairwise Independence
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Gowers Uniformity, Influence of Variables, and PCPs
SIAM Journal on Computing
SDP Integrality Gaps with Local ell_1-Embeddability
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Detecting high log-densities: an O(n¼) approximation for densest k-subgraph
Proceedings of the forty-second ACM symposium on Theory of computing
Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Dense subgraphs on dynamic networks
DISC'12 Proceedings of the 26th international conference on Distributed Computing
Mining structural hole spanners through information diffusion in social networks
Proceedings of the 22nd international conference on World Wide Web
Statistical algorithms and a lower bound for detecting planted cliques
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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The Densest k-subgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest k-subgraph: the current best algorithm gives an ≈ O(n1/4) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P ≠ NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest k-subgraph and its variants. Thus, understanding the approximability of Densest k-subgraph is an important challenge. In this work, we give evidence for the hardness of approximating Densest k-subgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest k-subgraph. Our results include: • A lower bound of Ω(n1/4/log3 n) on the integrality gap for Ω(log n/log log n) rounds of the Sherali-Adams relaxation for Densest k-subgraph. This also holds for the relaxation obtained from Sherali-Adams with an added SDP constraint. Our gap instances are in fact Erdös-Renyi random graphs. • For every ε 0, a lower bound of n2/53−ε on the integrality gap of nΩ(ε) rounds of the Lasserre SDP relaxation for Densest k-subgraph, and an nΩε(1) gap for n1−ε rounds. Our construction proceeds via a reduction from random instances of a certain Max-CSP over large domains. In the absence of inapproximability results for Densest k-subgraph, our results show that beating a factor of nΩ(1) is a barrier for even the most powerful SDPs, and in fact even beating the best known n1/4 factor is a barrier for current techniques. Our 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 nεΩ(1) rounds of the Lasserre hierarchy, where ε is the completeness parameter in Unique Games and Small Set Expansion.