Hardness of Max-2Lin and Max-3Lin over integers, reals, and large cyclic groups

Ryan O'Donnell, Yi Wu, Yuan Zhou

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

Abstract

In 1997, Håstad showed NP-hardness of (1 - ε,1/q + δ)-approximating Max-3Lin(ℤq); however it was not until 2007 that Guruswami and Raghavendra were able to show NP-hardness of (1 - ε,δ)-approximating Max-3Lin(ℤ). In 2004, Khot-Kindler-Mossel- O'Donnell showed UG-hardness of (1 - ε,δ)-approximating Max-2Lin(ℤq) for q = q(ε,δ) a sufficiently large constant; however achieving the same hardness for Max-2Lin(ℤ) was given as an open problem in Raghavendra's 2009 thesis. In this work we show that fairly simple modifications to the proofs of the Max-3Lin(ℤq) and Max-2Lin(ℤq) results yield optimal hardness results over ℤ. In fact, we show a kind of "bicriteria" hardness: even when there is a (1 - ε)-good solution over ℤ, it is hard for an algorithm to find a δ-good solution over ℤ, ℤ, or ℤm for any m ≥ q(ε, δ) of the algorithm's choosing.

Original languageEnglish (US)
Title of host publicationProceedings - 26th Annual IEEE Conference on Computational Complexity, CCC 2011
Pages23-33
Number of pages11
DOIs
StatePublished - Aug 29 2011
Externally publishedYes
Event26th Annual IEEE Conference on Computational Complexity, CCC 2011 - San Jose, CA, United States
Duration: Jun 8 2011Jun 10 2011

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159

Other

Other26th Annual IEEE Conference on Computational Complexity, CCC 2011
CountryUnited States
CitySan Jose, CA
Period6/8/116/10/11

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Computational Mathematics

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    O'Donnell, R., Wu, Y., & Zhou, Y. (2011). Hardness of Max-2Lin and Max-3Lin over integers, reals, and large cyclic groups. In Proceedings - 26th Annual IEEE Conference on Computational Complexity, CCC 2011 (pp. 23-33). [5959818] (Proceedings of the Annual IEEE Conference on Computational Complexity). https://doi.org/10.1109/CCC.2011.37