### Abstract

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 (^{1}_{2} − 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 (^{1}_{2} − 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 _{log}^{d}_{2 d} , where d is a constant. 2. NP-hardness of approximate the Maximum Acyclic Subgraph problem within a factor of ^{2}_{3} + ε, improving the previous ratio of ^{14}_{15} + 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 ^{1}_{2} − ε, it is NP-hard to satisfy more than ^{|}P−^{1(1)|} + ε fraction of constraints. q^{k.}

Original language | English (US) |
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Title of host publication | 34th Computational Complexity Conference, CCC 2019 |

Editors | Amir Shpilka |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771160 |

DOIs | |

State | Published - Jul 1 2019 |

Event | 34th Computational Complexity Conference, CCC 2019 - New Brunswick, United States Duration: Jul 18 2019 → Jul 20 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 137 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 34th Computational Complexity Conference, CCC 2019 |
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Country | United States |

City | New Brunswick |

Period | 7/18/19 → 7/20/19 |

### Fingerprint

### Keywords

- Inapproximability
- NP-hardness
- Unique games conjecture

### ASJC Scopus subject areas

- Software

### Cite this

*34th Computational Complexity Conference, CCC 2019*(Leibniz International Proceedings in Informatics, LIPIcs; Vol. 137). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.CCC.2019.3

**UG-hardness to NP-hardness by losing half.** / Bhangale, Amey; Khot, Subhash.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*34th Computational Complexity Conference, CCC 2019.*Leibniz International Proceedings in Informatics, LIPIcs, vol. 137, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 34th Computational Complexity Conference, CCC 2019, New Brunswick, United States, 7/18/19. https://doi.org/10.4230/LIPIcs.CCC.2019.3

}

TY - GEN

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

AU - Bhangale, Amey

AU - Khot, Subhash

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

UR - http://www.scopus.com/inward/record.url?scp=85070724182&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85070724182&partnerID=8YFLogxK

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

ER -