Improved NP-inapproximability for 2-variable linear equations

Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O'Donnell and John Wright

An instance of the 2-Lin(2) problem is a system of equations of the form . 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 fraction of the equations, for any (and any ). The previous best result, standing for over 15 years, had in place of . 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 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 .

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 greater than . Previously, no such limitation on gadget reductions was known.