### Abstract

We show an optimal hardness result for the following problem : Given a system of homogeneous linear equations over GF(2) with 3 variables per equation, find a balanced assignment that satisfies maximum number of equations. For arbitrarily small constant ζ > 0, we show that it is hard to determine (in polynomial time) whether such a system has a balanced assignment that satisfies 1 - ζ fraction of equations or there is no balanced assignment that satisfies more than 1/2 + ζ fraction of equations, As a corollary, we show that it is hard to approximate (in polynomial time) the Max-Bisection problem within factor 16/15 - ζ These hardness results hold under the assumption NP ⊈ > 0 DTIME(2 ^{nε}). Our results are obtained via a construction of a new PCP outer verifier that has a mixing property and a smoothness property. These properties are crucial in the analysis of the inner verifier. No previous outer verifier can achieve both these properties simultaneously. An outer verifier is essentially a 2-query PCP over a large alphabet. Loosely speaking, the mixing property says that the locations of the two queries read by the verifier are uncorrelated. The smoothness property says that the verifier's acceptance predicate is close to being a bijective predicate. Our construction relies on the algebraic techniques used to prove the PCP Theorem. This is in contrast with all earlier constructions that use the PCP Theorem as a black-box. The progress in inapproximability theory seems to require new ideas for building outer verifiers and our construction takes a first step in that direction.

Original language | English (US) |
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Title of host publication | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

Pages | 11-20 |

Number of pages | 10 |

State | Published - 2004 |

Event | Proceedings of the 36th Annual ACM Symposium on Theory of Computing - Chicago, IL, United States Duration: Jun 13 2004 → Jun 15 2004 |

### Other

Other | Proceedings of the 36th Annual ACM Symposium on Theory of Computing |
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Country | United States |

City | Chicago, IL |

Period | 6/13/04 → 6/15/04 |

### Fingerprint

### Keywords

- Hardness of Approximation
- Linear Equations
- Max-Bisection
- PCPs

### ASJC Scopus subject areas

- Software

### Cite this

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*(pp. 11-20)

**A new PCP outer verifier with applications to homogeneous linear equations and Max-Bisection.** / Holmerin, Jonas; Khot, Subhash.

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

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing.*pp. 11-20, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, United States, 6/13/04.

}

TY - GEN

T1 - A new PCP outer verifier with applications to homogeneous linear equations and Max-Bisection

AU - Holmerin, Jonas

AU - Khot, Subhash

PY - 2004

Y1 - 2004

N2 - We show an optimal hardness result for the following problem : Given a system of homogeneous linear equations over GF(2) with 3 variables per equation, find a balanced assignment that satisfies maximum number of equations. For arbitrarily small constant ζ > 0, we show that it is hard to determine (in polynomial time) whether such a system has a balanced assignment that satisfies 1 - ζ fraction of equations or there is no balanced assignment that satisfies more than 1/2 + ζ fraction of equations, As a corollary, we show that it is hard to approximate (in polynomial time) the Max-Bisection problem within factor 16/15 - ζ These hardness results hold under the assumption NP ⊈ > 0 DTIME(2 nε). Our results are obtained via a construction of a new PCP outer verifier that has a mixing property and a smoothness property. These properties are crucial in the analysis of the inner verifier. No previous outer verifier can achieve both these properties simultaneously. An outer verifier is essentially a 2-query PCP over a large alphabet. Loosely speaking, the mixing property says that the locations of the two queries read by the verifier are uncorrelated. The smoothness property says that the verifier's acceptance predicate is close to being a bijective predicate. Our construction relies on the algebraic techniques used to prove the PCP Theorem. This is in contrast with all earlier constructions that use the PCP Theorem as a black-box. The progress in inapproximability theory seems to require new ideas for building outer verifiers and our construction takes a first step in that direction.

AB - We show an optimal hardness result for the following problem : Given a system of homogeneous linear equations over GF(2) with 3 variables per equation, find a balanced assignment that satisfies maximum number of equations. For arbitrarily small constant ζ > 0, we show that it is hard to determine (in polynomial time) whether such a system has a balanced assignment that satisfies 1 - ζ fraction of equations or there is no balanced assignment that satisfies more than 1/2 + ζ fraction of equations, As a corollary, we show that it is hard to approximate (in polynomial time) the Max-Bisection problem within factor 16/15 - ζ These hardness results hold under the assumption NP ⊈ > 0 DTIME(2 nε). Our results are obtained via a construction of a new PCP outer verifier that has a mixing property and a smoothness property. These properties are crucial in the analysis of the inner verifier. No previous outer verifier can achieve both these properties simultaneously. An outer verifier is essentially a 2-query PCP over a large alphabet. Loosely speaking, the mixing property says that the locations of the two queries read by the verifier are uncorrelated. The smoothness property says that the verifier's acceptance predicate is close to being a bijective predicate. Our construction relies on the algebraic techniques used to prove the PCP Theorem. This is in contrast with all earlier constructions that use the PCP Theorem as a black-box. The progress in inapproximability theory seems to require new ideas for building outer verifiers and our construction takes a first step in that direction.

KW - Hardness of Approximation

KW - Linear Equations

KW - Max-Bisection

KW - PCPs

UR - http://www.scopus.com/inward/record.url?scp=4544354754&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4544354754&partnerID=8YFLogxK

M3 - Conference contribution

SP - 11

EP - 20

BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

ER -