### Abstract

We prove an improved hardness of approximation result for two problems, namely, the problem of finding the size of the largest clique in a graph and the problem of finding the chromatic number of a graph. We show that for any constant γ > 0, there is no polynomial time algorithm that approximates these problems within factor n/2^{(log n)3/4+γ} in an n vertex graph, assuming NP ⊈ BPTIME(2^{(log n)o(1)}). This improves the hardness factor of n/2^{(log n)1-γ′} for some small (unspecified) constant γ′ > 0 shown by Knot [20]. Our main idea is to show an improved hardness result for the Min-3Lin-Deletion problem. An instance of Min-3Lin-Deletion is a system of linear equations modulo 2, where each equation is over three variables. The objective is to find the minimum number of equations that need to be deleted so that the remaining system of equations has a satisfying assignment. We show a hardness factor of 2 ^{Ω(√log n)} for this problem, improving upon the hardness factor of (log n)^{β} shown by Håstad [18], for some small (unspecified) constant β > 0. The hardness results for clique and chromatic number are then obtained using the reduction from Min-3Lin-Deletion as given in [20].

Original language | English (US) |
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Title of host publication | Automata, Languages and Programming - 33rd International Colloquium, ICALP 2006, Proceedings |

Pages | 226-237 |

Number of pages | 12 |

Volume | 4051 LNCS |

DOIs | |

State | Published - 2006 |

Event | 33rd International Colloquium on Automata, Languages and Programming, ICALP 2006 - Venice, Italy Duration: Jul 10 2006 → Jul 14 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4051 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 33rd International Colloquium on Automata, Languages and Programming, ICALP 2006 |
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Country | Italy |

City | Venice |

Period | 7/10/06 → 7/14/06 |

### Fingerprint

### ASJC Scopus subject areas

- Biochemistry, Genetics and Molecular Biology(all)
- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Automata, Languages and Programming - 33rd International Colloquium, ICALP 2006, Proceedings*(Vol. 4051 LNCS, pp. 226-237). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4051 LNCS). https://doi.org/10.1007/11786986_21

**Better inapproximability results for maxclique, chromatic number and Min-3Lin-Deletion.** / Khot, Subhash; Ponnuswami, Ashok Kumar.

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

*Automata, Languages and Programming - 33rd International Colloquium, ICALP 2006, Proceedings.*vol. 4051 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4051 LNCS, pp. 226-237, 33rd International Colloquium on Automata, Languages and Programming, ICALP 2006, Venice, Italy, 7/10/06. https://doi.org/10.1007/11786986_21

}

TY - GEN

T1 - Better inapproximability results for maxclique, chromatic number and Min-3Lin-Deletion

AU - Khot, Subhash

AU - Ponnuswami, Ashok Kumar

PY - 2006

Y1 - 2006

N2 - We prove an improved hardness of approximation result for two problems, namely, the problem of finding the size of the largest clique in a graph and the problem of finding the chromatic number of a graph. We show that for any constant γ > 0, there is no polynomial time algorithm that approximates these problems within factor n/2(log n)3/4+γ in an n vertex graph, assuming NP ⊈ BPTIME(2(log n)o(1)). This improves the hardness factor of n/2(log n)1-γ′ for some small (unspecified) constant γ′ > 0 shown by Knot [20]. Our main idea is to show an improved hardness result for the Min-3Lin-Deletion problem. An instance of Min-3Lin-Deletion is a system of linear equations modulo 2, where each equation is over three variables. The objective is to find the minimum number of equations that need to be deleted so that the remaining system of equations has a satisfying assignment. We show a hardness factor of 2 Ω(√log n) for this problem, improving upon the hardness factor of (log n)β shown by Håstad [18], for some small (unspecified) constant β > 0. The hardness results for clique and chromatic number are then obtained using the reduction from Min-3Lin-Deletion as given in [20].

AB - We prove an improved hardness of approximation result for two problems, namely, the problem of finding the size of the largest clique in a graph and the problem of finding the chromatic number of a graph. We show that for any constant γ > 0, there is no polynomial time algorithm that approximates these problems within factor n/2(log n)3/4+γ in an n vertex graph, assuming NP ⊈ BPTIME(2(log n)o(1)). This improves the hardness factor of n/2(log n)1-γ′ for some small (unspecified) constant γ′ > 0 shown by Knot [20]. Our main idea is to show an improved hardness result for the Min-3Lin-Deletion problem. An instance of Min-3Lin-Deletion is a system of linear equations modulo 2, where each equation is over three variables. The objective is to find the minimum number of equations that need to be deleted so that the remaining system of equations has a satisfying assignment. We show a hardness factor of 2 Ω(√log n) for this problem, improving upon the hardness factor of (log n)β shown by Håstad [18], for some small (unspecified) constant β > 0. The hardness results for clique and chromatic number are then obtained using the reduction from Min-3Lin-Deletion as given in [20].

UR - http://www.scopus.com/inward/record.url?scp=33746347703&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33746347703&partnerID=8YFLogxK

U2 - 10.1007/11786986_21

DO - 10.1007/11786986_21

M3 - Conference contribution

AN - SCOPUS:33746347703

SN - 3540359044

SN - 9783540359043

VL - 4051 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 226

EP - 237

BT - Automata, Languages and Programming - 33rd International Colloquium, ICALP 2006, Proceedings

ER -