Approximation algorithms for semi-random partitioning problems
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Sorting noisy data with partial information
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Multireference alignment using semidefinite programming
Proceedings of the 5th conference on Innovations in theoretical computer science
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In this paper, we study the average case complexity of the Unique Games problem. We propose a semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Unique Games, then she chooses an epsilon-fraction of all edges, and finally replaces (& quot; corrupts'') the constraints corresponding to these edges with new constraints. If all steps are adversarial, the adversary can obtain any (1-epsilon)-satisfiable instance, so then the problem is as hard as in the worst case. We show however that we can find a solution satisfying a (1-delta) fraction of all constraints in polynomial-time if at least one step is random (we require that the average degree of the graph is Omeg(log k)). Our result holds only for epsilon less than some absolute constant. We prove that if epsilon = 1/2, then the problem is hard in one of the models, that is, no polynomial-time algorithm can distinguish between the following two cases: (i) the instance is a (1-epsilon)-satisfiable semi-random instance and (ii) the instance is at most delta-satisfiable (for every delta >, 0); the result assumes the 2-to-2 conjecture. Finally, we study semi-random instances of Unique Games that are at most (1-epsilon)-satisfiable. We present an algorithm that distinguishes between the case when the instance is a semi-random instance and the case when the instance is an (arbitrary) (1-delta)-satisfiable instances if epsilon >, c delta (for some absolute constant c).