How to Play Unique Games against a Semi-Random Adversary

In this paper, we study the average case complexity of the Unique Games problem. We propose a natural 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 ε–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 − ε) satisfiable instance, so then the problem is as hard as in the worst case. In our semi-random model, one of the steps is random, and all other steps are adversarial. We show that known algorithms for unique games (in particular, all algorithms that use the standard SDP relaxation) fail to solve semi-random instances of Unique Games. We present an algorithm that with high probability finds a solution satisfying a (1 − δ) fraction of all constraints in semi-random instances (we require that the average d...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Alexandra Kolla, Konstantin Makarychev, Yury Makarychev
• Keywords: Algorithm, CORR 2011, Education, Semi-random Instance, Unique Games