Interactive proofs and the hardness of approximating cliques
Journal of the ACM (JACM)
Approximating the independence number via the j -function
Mathematical Programming: Series A and B
The Lovász Theta Function and a Semidefinite Programming Relaxation of Vertex Cover
SIAM Journal on Discrete Mathematics
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On semidefinite programming relaxations for graph coloring and vertex cover
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs
SIAM Journal on Computing
An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
New Lower Bounds for Vertex Cover in the Lovasz-Schrijver Hierarchy
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Tight integrality gaps for Lovasz-Schrijver LP relaxations of vertex cover and max cut
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
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
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on 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
Expander flows, geometric embeddings and graph partitioning
Journal of the ACM (JACM)
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
A better approximation ratio for the vertex cover problem
ACM Transactions on Algorithms (TALG)
Vertex cover resists SDPs tightened by local hypermetric inequalities
IPCO'08 Proceedings of the 13th international conference on Integer programming and combinatorial optimization
How well can primal-dual and local-ratio algorithms perform?
ACM Transactions on Algorithms (TALG)
Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Exponential Lower Bounds and Integrality Gaps for Tree-Like Lovász-Schrijver Procedures
SIAM Journal on Computing
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Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NP-hard optimization problems. For example, breakthrough approximation algorithms for Max Cut and Sparsest Cut use semidefinite programming. Perhaps the most prominent NP-hard problem whose exact approximation factor is still unresolved is Vertex Cover. Probabilistically checkable proof (PCP)-based techniques of Dinur and Safra [Ann. of Math./ (2), 162 (2005), pp. 439-486] show that it is not possible to achieve a factor better than 1.36; on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm. There is a widespread belief that semidefinite programming (SDP) techniques are the most promising methods available for improving upon this factor of 2. Following a line of study initiated by Arora et al. [Theory Comput., 2 (2006), pp. 19-51], our aim is to show that a large family of linear programming (LP)- and SDP-based algorithms fail to produce an approximation for Vertex Cover better than 2. Lovász and Schrijver [SIAM J. Optim., 1 (1991), pp. 166-190] introduced the systems $LS$ and $LS_+$ for systematically tightening LP and SDP relaxations, respectively, over many rounds. These systems naturally capture large classes of LP and SDP relaxations; indeed, $LS_+$ captures the celebrated SDP-based algorithms for Max Cut and Sparsest Cut mentioned above. We rule out polynomial-time SDP-based $2-\Omega(1)$ approximations for Vertex Cover using $LS_+$. In particular, for every $\epsilon0$ we prove an integrality gap of $2-\epsilon$ for Vertex Cover SDPs obtained by tightening the standard LP relaxation with $\Omega(\sqrt{\log n/\log\log n})$ rounds of $LS_+$. While tight integrality gaps were known for Vertex Cover in the weaker $LS$ system [G. Schoenebeck, L. Trevisan, and M. Tulsiani, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2007, pp. 302-310], previous results did not rule out a $2-\Omega(1)$ approximation after even two rounds of $LS_+$.