Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
Proceedings of the forty-third annual ACM symposium on Theory of computing
Approximating CSPs with global cardinality constraints using SDP hierarchies
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
The complexity of conservative valued CSPs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Subsampling mathematical relaxations and average-case complexity
Proceedings of the twenty-second 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
A characterization of approximation resistance for even k-partite CSPs
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
The complexity of conservative valued CSPs
Journal of the ACM (JACM)
Proceedings of the 5th conference on Innovations in theoretical computer science
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A large number of interesting combinatorial optimization problems like MAX CUT, MAX k-SAT, and UNIQUE GAMES fall under the class of constraint satisfaction problems (CSPs). Recent work by one of the authors (STOC 2008) identifies a semidefinite programming (SDP) relaxation that yields the optimal approximation ratio for every CSP, under the Unique Games Conjecture (UGC). Very recently (FOCS 2009), the authors also showed unconditionally that the integrality gap of this basic SDP relaxation cannot be reduced by adding large classes of valid inequalities (e.g., in the fashion of Sherali--Adams LP hierarchies). In this work, we present an efficient rounding scheme that achieves the integrality gap of this basic SDP relaxation for every CSP (and it also achieves the gap of much stronger SDP relaxations). The SDP relaxation we consider is stronger or equivalent to any relaxation used in literature to approximate CSPs. Thus, irrespective of the truth of the UGC, our work yields an efficient generic algorithm that for every CSP, achieves an approximation at least as good as the best known algorithm in literature. The rounding algorithm in this paper can be summarized succinctly as follows: Reduce the dimension of SDP solution by random projection, discretize the projected vectors, and solve the resulting CSP instance by brute force! Even the proof is simple in that it avoids the use of the machinery from unique games reductions such as dictatorship tests, Fourier analysis or the invariance principle. A common theme of this paper and the subsequent paper in the same conference is a robustness lemma for SDP relaxations which asserts that approximately feasible solutions can be made feasible by "smoothing'' without changing the objective value significantly.