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Improved NP-Inapproximability for 2-Variable Linear Equations

Authors Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O’Donnell, John Wright

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Johan Håstad
Sangxia Huang
Rajsekar Manokaran
Ryan O’Donnell
John Wright

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Johan Håstad, Sangxia Huang, Rajsekar Manokaran, Ryan O’Donnell, and John Wright. Improved NP-Inapproximability for 2-Variable Linear Equations. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 341-360, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.341

Abstract

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 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 < 11/8 = 1.375 (and any 0 < epsilon <= 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.

Subject Classification

Keywords
  • approximability
  • unique games
  • linear equation
  • gadget
  • linear programming

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