### Abstract

In this paper we show that for n-vertex graphs with maximum degree 3, and for any fixed ε> 0, it is NP-hard to find α-edge separators and α-vertex separators of size no more than OPT+n^{ 1 2 - ε}, where OPT is the size of the optimal solutio n. For general graphs we show that it is NP-hard to find an α-edge separator of size no more than OPT+n^{2 - ε}. We also show that an O(f{hook}(n))-approximation algorithm for finding α-vertex separators of maximum degree 3 graphs can be used to find an O(f{hook}(n^{5}))-approximation algorithm for finding α-edge separators of general graphs. Since it is NP-hard to find optimal α-edge separators for general graphs this means that it is NP-hard to find optimal vertex separators even when restricted to maximum degree 3 graphs.

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

Pages (from-to) | 153-159 |

Number of pages | 7 |

Journal | Information Processing Letters |

Volume | 42 |

Issue number | 3 |

DOIs | |

State | Published - May 25 1992 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Signal Processing
- Information Systems
- Computer Science Applications

### Cite this

*Information Processing Letters*,

*42*(3), 153-159. https://doi.org/10.1016/0020-0190(92)90140-Q

}

*Information Processing Letters*, vol. 42, no. 3, pp. 153-159. https://doi.org/10.1016/0020-0190(92)90140-Q

**Finding good approximate vertex and edge partitions is NP-hard.** / Bui, Thang Nguyen; Jones, Curt.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Finding good approximate vertex and edge partitions is NP-hard

AU - Bui, Thang Nguyen

AU - Jones, Curt

PY - 1992/5/25

Y1 - 1992/5/25

N2 - In this paper we show that for n-vertex graphs with maximum degree 3, and for any fixed ε> 0, it is NP-hard to find α-edge separators and α-vertex separators of size no more than OPT+n 1 2 - ε, where OPT is the size of the optimal solutio n. For general graphs we show that it is NP-hard to find an α-edge separator of size no more than OPT+n2 - ε. We also show that an O(f{hook}(n))-approximation algorithm for finding α-vertex separators of maximum degree 3 graphs can be used to find an O(f{hook}(n5))-approximation algorithm for finding α-edge separators of general graphs. Since it is NP-hard to find optimal α-edge separators for general graphs this means that it is NP-hard to find optimal vertex separators even when restricted to maximum degree 3 graphs.

AB - In this paper we show that for n-vertex graphs with maximum degree 3, and for any fixed ε> 0, it is NP-hard to find α-edge separators and α-vertex separators of size no more than OPT+n 1 2 - ε, where OPT is the size of the optimal solutio n. For general graphs we show that it is NP-hard to find an α-edge separator of size no more than OPT+n2 - ε. We also show that an O(f{hook}(n))-approximation algorithm for finding α-vertex separators of maximum degree 3 graphs can be used to find an O(f{hook}(n5))-approximation algorithm for finding α-edge separators of general graphs. Since it is NP-hard to find optimal α-edge separators for general graphs this means that it is NP-hard to find optimal vertex separators even when restricted to maximum degree 3 graphs.

UR - http://www.scopus.com/inward/record.url?scp=0000301477&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000301477&partnerID=8YFLogxK

U2 - 10.1016/0020-0190(92)90140-Q

DO - 10.1016/0020-0190(92)90140-Q

M3 - Article

AN - SCOPUS:0000301477

VL - 42

SP - 153

EP - 159

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 3

ER -