Theory of Computing
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Title : Improved NP-Inapproximability for $2$-Variable Linear Equations
Authors : Johan Haastad, Sangxia Huang, Rajsekar Manokaran, Ryan O'Donnell, and John Wright
Volume : 13
Number : 19
Pages : 1-51
URL : https://theoryofcomputing.org/articles/v013a019
Abstract
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An instance of the 2-Lin problem is a system of equations of the
form "x_i + x_j = b mod 2." Given such a system in which it is
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
equations, for any C < 11/8 = 1.375 (and any 0 < \epsilon \leq 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 limitations on gadget reductions was known.
A preliminary version of this paper appeared in the
Proceedings of the 18th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems
(APPROX 2015).