Improved 3LIN hardness via linear label cover

Prahladh Harsha, Subhash Khot, Euiwoong Lee, Devanathan Thiruvenkatachari

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

Abstract

We prove that for every constant c and ε = (log n)−c, there is no polynomial time algorithm that when given an instance of 3-LIN with n variables where an (1 − ε)-fraction of the clauses are satisfiable, finds an assignment that satisfies atleast (1/2 + ε)-fraction of clauses unless NP ⊆ BPP. The previous best hardness using a polynomial time reduction achieves ε = (log log n)−c, which is obtained by the Label Cover hardness of Moshkovitz and Raz [J. ACM, 57(5), 2010] followed by the reduction from Label Cover to 3-LIN of Håstad [J. ACM, 48(4):798–859, 2001]. Our main idea is to prove a hardness result for Label Cover similar to Moshkovitz and Raz where each projection has a linear structure. This linear structure of Label Cover allows us to use Hadamard codes instead of long codes, making the reduction more efficient. For the hardness of Linear Label Cover, we follow the work of Dinur and Harsha [SIAM J. Comput., 42(6):2452–2486, 2013] that simplified the construction of Moshkovitz and Raz, and observe that running their reduction from a hardness of the problem LIN (of unbounded arity) instead of the more standard problem of solving quadratic equations ensures the linearity of the resultant Label Cover.

Original languageEnglish (US)
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019
EditorsDimitris Achlioptas, Laszlo A. Vegh
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771252
DOIs
StatePublished - Sep 2019
Event22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 - Cambridge, United States
Duration: Sep 20 2019Sep 22 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume145
ISSN (Print)1868-8969

Conference

Conference22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019
CountryUnited States
CityCambridge
Period9/20/199/22/19

Fingerprint

Labels
Hardness
Polynomials

Keywords

  • 3LIN
  • Composition
  • Low soundness error
  • PCP
  • Probabilistically checkable proofs

ASJC Scopus subject areas

  • Software

Cite this

Harsha, P., Khot, S., Lee, E., & Thiruvenkatachari, D. (2019). Improved 3LIN hardness via linear label cover. In D. Achlioptas, & L. A. Vegh (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019 [9] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 145). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.9

Improved 3LIN hardness via linear label cover. / Harsha, Prahladh; Khot, Subhash; Lee, Euiwoong; Thiruvenkatachari, Devanathan.

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019. ed. / Dimitris Achlioptas; Laszlo A. Vegh. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. 9 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 145).

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

Harsha, P, Khot, S, Lee, E & Thiruvenkatachari, D 2019, Improved 3LIN hardness via linear label cover. in D Achlioptas & LA Vegh (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019., 9, Leibniz International Proceedings in Informatics, LIPIcs, vol. 145, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019, Cambridge, United States, 9/20/19. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.9
Harsha P, Khot S, Lee E, Thiruvenkatachari D. Improved 3LIN hardness via linear label cover. In Achlioptas D, Vegh LA, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2019. 9. (Leibniz International Proceedings in Informatics, LIPIcs). https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.9
Harsha, Prahladh ; Khot, Subhash ; Lee, Euiwoong ; Thiruvenkatachari, Devanathan. / Improved 3LIN hardness via linear label cover. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019. editor / Dimitris Achlioptas ; Laszlo A. Vegh. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. (Leibniz International Proceedings in Informatics, LIPIcs).
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