## Abstract

In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of Bǎdoiu et al. (Proceedings of the eighteenth annual ACM-SIAM symposium on discrete algorithms (SODA 2007), pp. 512-521, 2007) and Bǎdoiu et al. (Proceedings of the 11th international workshop on approximation algorithms for combinatorial optimization problems (APPROX 2008), Springer, Berlin, pp. 21-34, 2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into an outerplanar metric. For this, we introduce a general notion of a metric relaxed minor and show that if G contains an α-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is ≥α. Then, for H=K_{2,3}, we present an algorithm which either finds an α-relaxed minor, or produces an O(α)-embedding into an outerplanar metric.

Original language | English |
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Pages (from-to) | 187-214 |

Number of pages | 28 |

Journal | Discrete and Computational Geometry |

Volume | 47 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2012 |

## Keywords

- Approximation algorithms
- Distortion
- Metric embedding
- Outerplanar graphs
- Trees

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics