Improved NP-inapproximability for 2-Variable Linear Equations *

Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O'donnell, John Wright
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 it's possible to satisfy all but an fraction of the equations, we show it is NP-hard to satisfy all but a C fraction of the equations, for any C < 11 8 = 1.375 (and any 0 < ≤ 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
more » ... ial 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. 1998 ACM Subject Classification F.2.0 Analysis of Algorithms and Problem Complexity