UG-Hardness to NP-Hardness by Losing Half

Authors Amey Bhangale, Subhash Khot

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Amey Bhangale
  • Weizmann Institute of Science, Rehovot, Israel
Subhash Khot
  • Courant Institute of Mathematical Sciences, New York University, USA


We would like to thank anonymous referees for helpful comments.

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Amey Bhangale and Subhash Khot. UG-Hardness to NP-Hardness by Losing Half. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • NP-hardness
  • Inapproximability
  • Unique Games Conjecture


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