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UG-Hardness to NP-Hardness by Losing Half

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Abstract

The 2-to-2 Games Theorem of [Subhash Khot et al., 2017; Dinur et al., 2018; Dinur et al., 2018; Dinur et al., 2018] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least (1/2-epsilon) fraction of the constraints vs. no assignment satisfying more than epsilon fraction of the constraints, for every constant epsilon>0. We show that the reduction can be transformed in a non-trivial way to give a stronger guarantee in the completeness case: For at least (1/2-epsilon) fraction of the vertices on one side, all the constraints associated with them in the Unique Games instance can be satisfied. We use this guarantee to convert the known UG-hardness results to NP-hardness. We show: 1) Tight inapproximability of approximating independent sets in degree d graphs within a factor of Omega(d/(log^2 d)), where d is a constant. 2) NP-hardness of approximate the Maximum Acyclic Subgraph problem within a factor of 2/3+epsilon, improving the previous ratio of 14/15+epsilon by Austrin et al. [Austrin et al., 2015]. 3) For any predicate P^{-1}(1) subseteq [q]^k supporting a balanced pairwise independent distribution, given a P-CSP instance with value at least 1/2-epsilon, it is NP-hard to satisfy more than (|P^{-1}(1)|/(q^k))+epsilon fraction of constraints.

BibTeX - Entry

@InProceedings{bhangale_et_al:LIPIcs:2019:10825,
  author =	{Amey Bhangale and Subhash Khot},
  title =	{{UG-Hardness to NP-Hardness by Losing Half}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{3:1--3:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Amir Shpilka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10825},
  URN =		{urn:nbn:de:0030-drops-108258},
  doi =		{10.4230/LIPIcs.CCC.2019.3},
  annote =	{Keywords: NP-hardness, Inapproximability, Unique Games Conjecture}
}

Keywords: NP-hardness, Inapproximability, Unique Games Conjecture
Seminar: 34th Computational Complexity Conference (CCC 2019)
Issue date: 2019
Date of publication: 2019


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