How to Play Unique Games against a Semi-Random Adversary Study of Semi-Random Models of Unique Games

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 ("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 (Omega) over tilde (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).