Ryan O’Donnell

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

Optimal Lower Bounds for Locality-Sensitive Hashing (Except When q is Tiny)

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

New degree bounds for polynomial threshold functions

for all

Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs

\epsilon > 0

Optimal Bounds for Estimating Entropy with PMF Queries

; here

Algorithmic Signaling of Features in Auction Design

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

A New Point of NP-Hardness for 2-to-1 Label Cover

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-

New NP-Hardness Results for 3-Coloring and 2-to-1 Label Cover

q

On the Fourier tails of bounded functions over the discrete cube

-CUT, and MAX-2LIN(

Hardness of Max-2Lin and Max-3Lin over Integers, Reals, and Large Cyclic Groups

q