Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
SDP gaps for 2-to-1 and other label-cover variants
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Approximating sparsest cut in graphs of bounded treewidth
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Approximate Lasserre integrality gap for unique games
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Improving integrality gaps via Chvátal-Gomory rounding
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Integrality gaps of linear and semi-definite programming relaxations for Knapsack
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
Approximating CSPs with global cardinality constraints using SDP hierarchies
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On LP-based approximability for strict CSPs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Semidefinite programming and approximation algorithms: a survey
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Hypercontractivity, sum-of-squares proofs, and their applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Sparsest cut on quotients of the hypercube
CATS '11 Proceedings of the Seventeenth Computing: The Australasian Theory Symposium - Volume 119
Sparsest cut on quotients of the hypercube
CATS 2011 Proceedings of the Seventeenth Computing on The Australasian Theory Symposium - Volume 119
A PRG for lipschitz functions of polynomials with applications to sparsest cut
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Sparsest cut on bounded treewidth graphs: algorithms and hardness results
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Majority is stablest: discrete and SoS
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Cones of multipowers and combinatorial optimization problems
Computational Mathematics and Mathematical Physics
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With the work of Khot and Vishnoi (FOCS 2005) as a starting point, we obtain integrality gaps for certain strong SDP relaxations of unique games. Specifically, we exhibit a gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to $\exp(\Omega(\log\log~n)^{1/4})$ vectors. For stronger relaxations obtained from the basic semidefinite program by $R$ rounds of Sherali--Adams lift-and-project, we prove a unique games integrality gap for $R = \Omega(\log\log~n)^{1/4}$.By composing these SDP gaps with UGC-hardness reductions, the above results imply corresponding integrality gaps for every problem for which a UGC-based hardness is known. Consequently, this work implies that including any valid constraints on up to$\exp(\Omega(\log\log~n)^{1/4})$ vectors to natural semidefinite program, does not improve the approximation ratio for any problem in the following classes: constraint satisfaction problems, ordering constraint satisfaction problems and metric labeling problems over constant-size metrics. We obtain similar SDP integrality gaps for balanced separator, building on Devanur et al. (STOC 2006). We also exhibit, for explicit constants $\gamma, \delta 0$, an n-point negative-type metric which requires distortion $\Omega(\log\log n)^{\gamma}$ to embed into$\ell_1$, although all its subsets of size$\exp(\Omega(\log\log~n)^{\delta})$ embed isometrically into $\ell_1$.