We give a subexponential-time approximation algorithm for the Unique Games problem: Given a Unique Games instance with optimal value \(1-\varepsilon^3\) and alphabet size \(k\), our algorithm finds in time \(\exp(k\cdot n^\varepsilon)\) a solution of constant value (say at least 0.99).
We also obtain subexponential algorithms with similar approximation guarantees for Small-Set Expansion and Multi Cut. For Max Cut, Sparsest Cut and Vertex Cover, our techniques lead to subexponential algorithms with improved approximation guarantees on interesting subclasses of instances.
Khot’s Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for Unique Games. While our result stops short of refuting the UGC, it does suggest that Unique Games is significantly easier than NP-hard problems such as Max 3-SAT, Label Cover and more, that are believed not to have subexponential algorithms achieving a non-trivial approximation ratio.
The main component in our algorithms is a new kind of graph decomposition that may have other applications: We show that every graph with \(n\) vertices can be efficiently partitioned into disjoint induced subgraphs, each with at most \(n^\varepsilon\) eigenvalues above \(1-\varepsilon^3\), such that at most 0.01 of the edges do not respect the partition.
Joint work with Sanjeev Arora and Boaz Barak.