### Abstract

We describe logarithmic approximation algorithms for the NP-hard graph optimization problems of minimum linear arrangement, minimum containing interval graph, and minimum storage-time product. This improves upon the best previous approximation bounds of Even, Naor, Rao, and Schieber [J. ACM, 47 (2000), pp. 585-616] for these problems by a factor of Ω(log log n). We use the lower bound provided by the volume W of a spreading metric for each of the ordering problems above (as defined by Even et al.) in order to find a solution with cost at most a logarithmic factor times W for these problems. We develop a divide-and-conquer strategy where the cost of a solution to a problem at a recursive level is C plus the cost of a solution to the subproblems at this level, and where the spreading metric volume on the subproblems is less than the original volume by Ω(C/ log n), ensuring that the resulting solution has cost O(log n) times the original spreading metric volume. We note that this is an existentially tight bound on the relationship between the spreading metric volume and the true optimal values for these problems. For planar graphs, we combine a structural theorem of Klein, Plotkin, and Rao [Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 682-690] with our new recursion technique to show that the spreading metric cost volumes are within an O(log log n) factor of the cost of an optimal solution for the minimum linear arrangement, and the minimum containing interval graph problems.

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

Pages (from-to) | 388-404 |

Number of pages | 17 |

Journal | SIAM Journal on Computing |

Volume | 34 |

Issue number | 2 |

DOIs | |

State | Published - 2005 |

### Fingerprint

### Keywords

- Approximation algorithms
- Interval graph completion
- Minimum linear arrangement
- Spreading metrics
- Storage - time product

### ASJC Scopus subject areas

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

### Cite this

*SIAM Journal on Computing*,

*34*(2), 388-404. https://doi.org/10.1137/S0097539702413197

**New approximation techniques for some linear ordering problems.** / Rao, Satish; Richa, Andrea.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 34, no. 2, pp. 388-404. https://doi.org/10.1137/S0097539702413197

}

TY - JOUR

T1 - New approximation techniques for some linear ordering problems

AU - Rao, Satish

AU - Richa, Andrea

PY - 2005

Y1 - 2005

N2 - We describe logarithmic approximation algorithms for the NP-hard graph optimization problems of minimum linear arrangement, minimum containing interval graph, and minimum storage-time product. This improves upon the best previous approximation bounds of Even, Naor, Rao, and Schieber [J. ACM, 47 (2000), pp. 585-616] for these problems by a factor of Ω(log log n). We use the lower bound provided by the volume W of a spreading metric for each of the ordering problems above (as defined by Even et al.) in order to find a solution with cost at most a logarithmic factor times W for these problems. We develop a divide-and-conquer strategy where the cost of a solution to a problem at a recursive level is C plus the cost of a solution to the subproblems at this level, and where the spreading metric volume on the subproblems is less than the original volume by Ω(C/ log n), ensuring that the resulting solution has cost O(log n) times the original spreading metric volume. We note that this is an existentially tight bound on the relationship between the spreading metric volume and the true optimal values for these problems. For planar graphs, we combine a structural theorem of Klein, Plotkin, and Rao [Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 682-690] with our new recursion technique to show that the spreading metric cost volumes are within an O(log log n) factor of the cost of an optimal solution for the minimum linear arrangement, and the minimum containing interval graph problems.

AB - We describe logarithmic approximation algorithms for the NP-hard graph optimization problems of minimum linear arrangement, minimum containing interval graph, and minimum storage-time product. This improves upon the best previous approximation bounds of Even, Naor, Rao, and Schieber [J. ACM, 47 (2000), pp. 585-616] for these problems by a factor of Ω(log log n). We use the lower bound provided by the volume W of a spreading metric for each of the ordering problems above (as defined by Even et al.) in order to find a solution with cost at most a logarithmic factor times W for these problems. We develop a divide-and-conquer strategy where the cost of a solution to a problem at a recursive level is C plus the cost of a solution to the subproblems at this level, and where the spreading metric volume on the subproblems is less than the original volume by Ω(C/ log n), ensuring that the resulting solution has cost O(log n) times the original spreading metric volume. We note that this is an existentially tight bound on the relationship between the spreading metric volume and the true optimal values for these problems. For planar graphs, we combine a structural theorem of Klein, Plotkin, and Rao [Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 682-690] with our new recursion technique to show that the spreading metric cost volumes are within an O(log log n) factor of the cost of an optimal solution for the minimum linear arrangement, and the minimum containing interval graph problems.

KW - Approximation algorithms

KW - Interval graph completion

KW - Minimum linear arrangement

KW - Spreading metrics

KW - Storage - time product

UR - http://www.scopus.com/inward/record.url?scp=18444402512&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18444402512&partnerID=8YFLogxK

U2 - 10.1137/S0097539702413197

DO - 10.1137/S0097539702413197

M3 - Article

AN - SCOPUS:18444402512

VL - 34

SP - 388

EP - 404

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 2

ER -