UG-hardness to NP-hardness by losing half

Amey Bhangale, Subhash Khot

Research output: Chapter in Book/Report/Conference proceedingConference contribution


The 2-to-2 Games Theorem of [16, 10, 11, 17] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least (12 − ε) fraction of the constraints vs. no assignment satisfying more than ε fraction of the constraints, for every constant ε > 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 (12 − ε) 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 Ω logd2 d , where d is a constant. 2. NP-hardness of approximate the Maximum Acyclic Subgraph problem within a factor of 23 + ε, improving the previous ratio of 1415 + ε by Austrin et al. [4]. 3. For any predicate P−1(1) ⊆ [q]k supporting a balanced pairwise independent distribution, given a P-CSP instance with value at least 12 − ε, it is NP-hard to satisfy more than |P−1(1)| + ε fraction of constraints. qk.

Original languageEnglish (US)
Title of host publication34th Computational Complexity Conference, CCC 2019
EditorsAmir Shpilka
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771160
StatePublished - Jul 1 2019
Event34th Computational Complexity Conference, CCC 2019 - New Brunswick, United States
Duration: Jul 18 2019Jul 20 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference34th Computational Complexity Conference, CCC 2019
Country/TerritoryUnited States
CityNew Brunswick


  • Inapproximability
  • NP-hardness
  • Unique games conjecture

ASJC Scopus subject areas

  • Software


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