Subexponential Algorithms for Unique Games and Related Problems

We give a sub exponential time approximation algorithm for the \textsc{Unique Games} problem. The algorithms run in time that is exponential in an arbitrarily small polynomial of the input size, $n^{\epsilon}$. The approximation guarantee depends on~$\epsilon$, but not on the alphabet size or the number of variables. We also obtain a sub exponential algorithms with improved approximations for \textsc{Small-Set Expansion} and \textsc{Multicut}. For \textsc{Max Cut}, \textsc{Sparsest Cut}, and \textsc{Vertex Cover}, we give sub exponential algorithms with improved approximations on some interesting subclasses of instances. Khot's Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for the \textsc{Unique Games}. While our results stop short of refuting the UGC, they do suggest that \textsc{Unique Games} is significantly easier than NP-hard problems such as \textsc{Max 3Sat}, \textsc{Max 3Lin}, \textsc{Label Cover} and more, that are believed not to have a sub exponential algorithm achieving a non-trivial approximation ratio. The main component in our algorithms is a new result on graph decomposition that may have other applications. Namely we show that for every $\epsilon0$ and every regular $n$-vertex graph~$G$, by changing at most $\epsilon$ fraction of $G$'s edges, one can break~$G$ into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most $ n^{\epsilon}$ eigenvalues larger than $1-\eta$, where $\eta$ depends polynomially on $\epsilon$.