Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Optimal algorithms and inapproximability results for every CSP?
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
An optimal sdp algorithm for max-cut, and equally optimal long code tests
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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
Exponential lower bounds and integrality gaps for tree-like Lovász-Schrijver procedures
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Near-optimal algorithms for maximum constraint satisfaction problems
ACM Transactions on Algorithms (TALG)
Improved rounding for parallel repeated 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
Exponential Lower Bounds and Integrality Gaps for Tree-Like Lovász-Schrijver Procedures
SIAM Journal on Computing
On quadratic programming with a ratio objective
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
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We continue the recent line of work on the connection between semidefinite programming-based approximation algorithms and the Unique Games Conjecture. Given any boolean 2-CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clause-by-clause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which are "hard" to round in a certain sense, we obtain a Unique Games-based inapproximability matching this "hardness" of rounding the set of vector triples. Conversely, any instance together with an SDP solution can be viewed as a set of vector triples, and we show that we can always find an assignment to the instance which is at least as good as the "hardness" of rounding the corresponding set of vector triples. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2-CSPs.As an application, we show that MAX 2-AND is hard to approximate within 0.87435. This improves upon the best previous hardness of \alpha GW +\in\approx 0.87856, and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of MAX 2-AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor \alpha GW.