### 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 language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019 |

Editors | Dimitris Achlioptas, Laszlo A. Vegh |

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

ISBN (Electronic) | 9783959771252 |

DOIs | |

State | Published - Sep 2019 |

Event | 22nd 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 2019 → Sep 22 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 145 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 |
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Country | United States |

City | Cambridge |

Period | 9/20/19 → 9/22/19 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Software

### Cite this

*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.

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

*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

}

TY - GEN

T1 - Improved 3LIN hardness via linear label cover

AU - Harsha, Prahladh

AU - Khot, Subhash

AU - Lee, Euiwoong

AU - Thiruvenkatachari, Devanathan

PY - 2019/9

Y1 - 2019/9

N2 - 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.

AB - 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.

KW - 3LIN

KW - Composition

KW - Low soundness error

KW - PCP

KW - Probabilistically checkable proofs

UR - http://www.scopus.com/inward/record.url?scp=85072851289&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072851289&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2019.9

DO - 10.4230/LIPIcs.APPROX-RANDOM.2019.9

M3 - Conference contribution

AN - SCOPUS:85072851289

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019

A2 - Achlioptas, Dimitris

A2 - Vegh, Laszlo A.

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

ER -