Subhash Khot

Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of

Ruling Out PTAS for Graph Min-Bisection, Dense k-Subgraph, and Bipartite Clique

\alpha_{\text{\tiny{GW}}} + \epsilon

Improved inapproximability results for MaxClique, chromatic number and approximate graph coloring

for all

Near-optimal lower bounds on the multi-party communication complexity of set disjointness

\epsilon > 0

Hardness of approximating the shortest vector problem in lattices

; here

A new multilayered PCP and the hardness of hypergraph vertex cover

\alpha_{\text{\tiny{GW}}} \approx .878567

New Results for Learning Noisy Parities and Halfspaces

denotes the approximation ratio achieved by the algorithm of Goemans and Williamson in [J. Assoc. Comput. Mach., 42 (1995), pp. 1115-1145]. This implies that if the Unique Games Conjecture of Khot in [Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767-775] holds, then the Goemans-Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21-30]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21-30] contains a proof of an asymptotic version of it. Our techniques extend to several other two-variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX-2SAT, MAX-

Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?

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Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique

-CUT, and MAX-2LIN(

Inapproximability results for combinatorial auctions with submodular utility functions

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