NP-hardness of approximately solving linear equations over reals
Proceedings of the forty-third annual ACM symposium on Theory of computing
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Algorithmic extensions of cheeger's inequality to higher eigenvalues and partitions
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Randomly Supported Independence and Resistance
SIAM Journal on Computing
Bypassing UGC from some optimal geometric inapproximability results
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
A new point of NP-hardness for unique games
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Hypercontractivity, sum-of-squares proofs, and their applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Multi-way spectral partitioning and higher-order cheeger inequalities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Many sparse cuts via higher eigenvalues
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
Optimizing password composition policies
Proceedings of the fourteenth ACM conference on Electronic commerce
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Approximation resistance from pairwise independent subgroups
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
New NP-Hardness Results for 3-Coloring and 2-to-1 Label Cover
ACM Transactions on Computation Theory (TOCT)
Robust Satisfiability for CSPs: Hardness and Algorithmic Results
ACM Transactions on Computation Theory (TOCT)
Locally testable codes and cayley graphs
Proceedings of the 5th conference on Innovations in theoretical computer science
On a connection between small set expansions and modularity clustering
Information Processing Letters
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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 $\epsilon$.