Improved NP-Inapproximability for 2-Variable Linear Equations.

An instance of the 2-Lin(2) problem is a system of equations of the form x_i + x_j = b (mod 2). Given such a system in which itu0027s possible to satisfy all but an epsilon fraction of the equations, we show it is NP-hard to satisfy all but a C*epsilon fraction of the equations, for any C u003c 11/8 = 1.375 (and any 0 u003c epsilon u003c= 1/8). The previous best result, standing for over 15 years, had 5/4 in place of 11/8. Our result provides the best known NP-hardness even for the Unique Games problem, and it also holds for the special case of Max-Cut. The precise factor 11/8 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 3/2.Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitation on gadget reductions was known.