### Abstract

Given a k-uniform hypergraph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyperedge. We present a new multilayered probabilistically checkable proof (PCP) construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within a factor of (k - 1 - ε) for arbitrary constants ε > 0 and k ≥ 3. The result is nearly tight as this problem can be easily approximated within factor k. Our construction makes use of the biased long-code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets. We also give a different proof that shows an inapproximability factor of [k/2] - ε. In addition to being simpler, this proof also works for superconstant values of k up to (log N) ^{1/c}, where c > 1 is a fixed constant and N is the number of hyperedges.

Original language | English (US) |
---|---|

Pages (from-to) | 1129-1146 |

Number of pages | 18 |

Journal | SIAM Journal on Computing |

Volume | 34 |

Issue number | 5 |

DOIs | |

State | Published - 2005 |

### Fingerprint

### Keywords

- Hardness of approximation
- Hypergraph vertex cover
- Long-code
- Multilayered outer verifier
- Probabilistically checkable proof

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*SIAM Journal on Computing*,

*34*(5), 1129-1146. https://doi.org/10.1137/S0097539704443057

**A new multilayered PCP and the hardness of hypergraph vertex cover.** / Dinur, Irit; Guruswami, Venkatesan; Khot, Subhash; Regev, Oded.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 34, no. 5, pp. 1129-1146. https://doi.org/10.1137/S0097539704443057

}

TY - JOUR

T1 - A new multilayered PCP and the hardness of hypergraph vertex cover

AU - Dinur, Irit

AU - Guruswami, Venkatesan

AU - Khot, Subhash

AU - Regev, Oded

PY - 2005

Y1 - 2005

N2 - Given a k-uniform hypergraph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyperedge. We present a new multilayered probabilistically checkable proof (PCP) construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within a factor of (k - 1 - ε) for arbitrary constants ε > 0 and k ≥ 3. The result is nearly tight as this problem can be easily approximated within factor k. Our construction makes use of the biased long-code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets. We also give a different proof that shows an inapproximability factor of [k/2] - ε. In addition to being simpler, this proof also works for superconstant values of k up to (log N) 1/c, where c > 1 is a fixed constant and N is the number of hyperedges.

AB - Given a k-uniform hypergraph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyperedge. We present a new multilayered probabilistically checkable proof (PCP) construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within a factor of (k - 1 - ε) for arbitrary constants ε > 0 and k ≥ 3. The result is nearly tight as this problem can be easily approximated within factor k. Our construction makes use of the biased long-code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets. We also give a different proof that shows an inapproximability factor of [k/2] - ε. In addition to being simpler, this proof also works for superconstant values of k up to (log N) 1/c, where c > 1 is a fixed constant and N is the number of hyperedges.

KW - Hardness of approximation

KW - Hypergraph vertex cover

KW - Long-code

KW - Multilayered outer verifier

KW - Probabilistically checkable proof

UR - http://www.scopus.com/inward/record.url?scp=27144509145&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=27144509145&partnerID=8YFLogxK

U2 - 10.1137/S0097539704443057

DO - 10.1137/S0097539704443057

M3 - Article

VL - 34

SP - 1129

EP - 1146

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 5

ER -