TY - GEN

T1 - UG-hardness to NP-hardness by losing half

AU - Bhangale, Amey

AU - Khot, Subhash

N1 - Funding Information:
Funding Amey Bhangale: Research supported by Irit Dinur’s ERC-CoG grant 772839. Subhash Khot: Research supported by NSF CCF-1813438, Simons Collaboration on Algorithms and Geometry, and Simons Investigator Award.

PY - 2019/7/1

Y1 - 2019/7/1

N2 - 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.

AB - 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.

KW - Inapproximability

KW - NP-hardness

KW - Unique games conjecture

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U2 - 10.4230/LIPIcs.CCC.2019.3

DO - 10.4230/LIPIcs.CCC.2019.3

M3 - Conference contribution

AN - SCOPUS:85070724182

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 34th Computational Complexity Conference, CCC 2019

A2 - Shpilka, Amir

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 34th Computational Complexity Conference, CCC 2019

Y2 - 18 July 2019 through 20 July 2019

ER -