Constant approximation algorithms for embedding graph metrics into trees and outerplanar graphs

Authors:
Victor Chepoi;Feodor F. Dragan;Ilan Newman;Yuri Rabinovich;Yann Vaxès
Affiliations:
LIF, Universitée d'Aix-Marseille, Marseille, France;Computer Science Department, Kent State University, Kent, OH;Department of Computer Science, University of Haifa, Haifa, Israel;Department of Computer Science, University of Haifa, Haifa, Israel;LIF, Universitée d'Aix-Marseille, Marseille, France
Venue:
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Year:
2010

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Abstract

We present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding (unweighted) graph metrics into tree metrics (thus improving and simplifying the factor 100 and 27 algorithms of Badoiu et al. (2007) and Badoiu et al. (2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding graph metrics into outerplanar metrics. For this, we introduce a notion of 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 = K2,3 we present an algorithm which either finds an α-relaxed minor, or produces an O(α)-embedding into an outerplanar metric.