eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-07-16
3:1
3:20
10.4230/LIPIcs.CCC.2019.3
article
UG-Hardness to NP-Hardness by Losing Half
Bhangale, Amey
1
Khot, Subhash
2
Weizmann Institute of Science, Rehovot, Israel
Courant Institute of Mathematical Sciences, New York University, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol137-ccc2019/LIPIcs.CCC.2019.3/LIPIcs.CCC.2019.3.pdf
NP-hardness
Inapproximability
Unique Games Conjecture