UG-hardness to NP-hardness by losing half

Amey Bhangale, Subhash Khot

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

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 (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
DOIs
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
Volume137
ISSN (Print)1868-8969

Conference

Conference34th Computational Complexity Conference, CCC 2019
CountryUnited States
CityNew Brunswick
Period7/18/197/20/19

Fingerprint

Hardness

Keywords

  • Inapproximability
  • NP-hardness
  • Unique games conjecture

ASJC Scopus subject areas

  • Software

Cite this

Bhangale, A., & Khot, S. (2019). UG-hardness to NP-hardness by losing half. In A. Shpilka (Ed.), 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.

34th Computational Complexity Conference, CCC 2019. ed. / Amir Shpilka. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 137).

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

Bhangale, A & Khot, S 2019, UG-hardness to NP-hardness by losing half. in A Shpilka (ed.), 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
Bhangale A, Khot S. UG-hardness to NP-hardness by losing half. In Shpilka A, editor, 34th Computational Complexity Conference, CCC 2019. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2019. (Leibniz International Proceedings in Informatics, LIPIcs). https://doi.org/10.4230/LIPIcs.CCC.2019.3
Bhangale, Amey ; Khot, Subhash. / UG-hardness to NP-hardness by losing half. 34th Computational Complexity Conference, CCC 2019. editor / Amir Shpilka. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. (Leibniz International Proceedings in Informatics, LIPIcs).
@inproceedings{74e46f002666418191dcdf29f75045e7,
title = "UG-hardness to NP-hardness by losing half",
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 (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.",
keywords = "Inapproximability, NP-hardness, Unique games conjecture",
author = "Amey Bhangale and Subhash Khot",
year = "2019",
month = "7",
day = "1",
doi = "10.4230/LIPIcs.CCC.2019.3",
language = "English (US)",
series = "Leibniz International Proceedings in Informatics, LIPIcs",
publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",
editor = "Amir Shpilka",
booktitle = "34th Computational Complexity Conference, CCC 2019",

}

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 -