### Abstract

Assuming that NP ⊈ ∩_{ε > 0} BPTIME(2 ^{nε}), we show that graph min-bisection, dense fc-subgraph, and bipartite clique have no polynomial time approximation scheme (PTAS). We give a reduction from the minimum distance of code (MDC) problem. Starting with an instance of MDC, we build a quasi-random probabilistically checkable proof (PCP) that suffices to prove the desired inapproximability results. In a quasi-random PCP, the query pattern of the verifier looks random in a certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial, and it can be checked probabilistically by reading a constant number of its values.

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

Pages (from-to) | 1025-1071 |

Number of pages | 47 |

Journal | SIAM Journal on Computing |

Volume | 36 |

Issue number | 4 |

DOIs | |

State | Published - 2006 |

### Fingerprint

### Keywords

- Approximation algorithms
- Hardness of approximation
- Probabilistically checkable proofs (PCPs)

### ASJC Scopus subject areas

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

### Cite this

**Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique.** / Khot, Subhash.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 36, no. 4, pp. 1025-1071. https://doi.org/10.1137/S0097539705447037

}

TY - JOUR

T1 - Ruling out ptas for graph min-bisection, dense k-subgraph, and bipartite clique

AU - Khot, Subhash

PY - 2006

Y1 - 2006

N2 - Assuming that NP ⊈ ∩ε > 0 BPTIME(2 nε), we show that graph min-bisection, dense fc-subgraph, and bipartite clique have no polynomial time approximation scheme (PTAS). We give a reduction from the minimum distance of code (MDC) problem. Starting with an instance of MDC, we build a quasi-random probabilistically checkable proof (PCP) that suffices to prove the desired inapproximability results. In a quasi-random PCP, the query pattern of the verifier looks random in a certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial, and it can be checked probabilistically by reading a constant number of its values.

AB - Assuming that NP ⊈ ∩ε > 0 BPTIME(2 nε), we show that graph min-bisection, dense fc-subgraph, and bipartite clique have no polynomial time approximation scheme (PTAS). We give a reduction from the minimum distance of code (MDC) problem. Starting with an instance of MDC, we build a quasi-random probabilistically checkable proof (PCP) that suffices to prove the desired inapproximability results. In a quasi-random PCP, the query pattern of the verifier looks random in a certain precise sense. Among the several new techniques we introduce, the most interesting one gives a way of certifying that a given polynomial belongs to a given linear subspace of polynomials. As is important for our purpose, the certificate itself happens to be another polynomial, and it can be checked probabilistically by reading a constant number of its values.

KW - Approximation algorithms

KW - Hardness of approximation

KW - Probabilistically checkable proofs (PCPs)

UR - http://www.scopus.com/inward/record.url?scp=34547839125&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547839125&partnerID=8YFLogxK

U2 - 10.1137/S0097539705447037

DO - 10.1137/S0097539705447037

M3 - Article

AN - SCOPUS:34547839125

VL - 36

SP - 1025

EP - 1071

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 4

ER -