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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.341
URN: urn:nbn:de:0030-drops-53112
URL: https://drops.dagstuhl.de/opus/volltexte/2015/5311/
Håstad, Johan ; Huang, Sangxia ; Manokaran, Rajsekar ; O’Donnell, Ryan ; Wright, John
Improved NP-Inapproximability for 2-Variable Linear Equations
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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.
BibTeX - Entry
@InProceedings{hstad_et_al:LIPIcs:2015:5311,
author = {Johan Håstad and Sangxia Huang and Rajsekar Manokaran and Ryan O’Donnell and John Wright},
title = {{Improved NP-Inapproximability for 2-Variable Linear Equations}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
pages = {341--360},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-89-7},
ISSN = {1868-8969},
year = {2015},
volume = {40},
editor = {Naveen Garg and Klaus Jansen and Anup Rao and Jos{\'e} D. P. Rolim},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5311},
URN = {urn:nbn:de:0030-drops-53112},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.341},
annote = {Keywords: approximability, unique games, linear equation, gadget, linear programming}
}
| Keywords: | approximability, unique games, linear equation, gadget, linear programming | |
| Seminar: | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015) | |
| Issue date: | 2015 | |
| Date of publication: | 13.08.2015 |
