Showing papers by "Yuri Rabinovich published in 2006"

Embedding k -Outerplanar Graphs into l 1

Abstract: We show that the shortest-path metric of any $k$-outerplanar graph, for any fixed $k$, can be approximated by a probability distribution over tree metrics with constant distortion and hence also embedded into $\ell_1$ with constant distortion. These graphs play a central role in polynomial time approximation schemes for many NP-hard optimization problems on general planar graphs and include the family of weighted $k\times n$ planar grids. This result implies a constant upper bound on the ratio between the sparsest cut and the maximum concurrent flow in multicommodity networks for $k$-outerplanar graphs, thus extending a theorem of Okamura and Seymour [J. Combin. Theory Ser. B, 31 (1981), pp. 75-81] for outerplanar graphs, and a result of Gupta et al. [Combinatorica, 24 (2004), pp. 233-269] for treewidth-2 graphs. In addition, we obtain improved approximation ratios for $k$-outerplanar graphs on various problems for which approximation algorithms are based on probabilistic tree embeddings. We conjecture that these embeddings for $k$-outerplanar graphs may serve as building blocks for $\ell_1$ embeddings of more general metrics.

Local versus global properties of metric spaces

Abstract: Motivated by applications in combinatorial optimization, we initiate a study of the extent to which the global properties of a metric space (especially, embeddability in l1 with low distortion) are determined by the properties of small subspaces. We note connections to similar issues studied already in Ramsey theory, complexity theory (especially PCPs), and property testing. We prove both upper bounds and lower bounds on the distortion of embedding locally constrained metrics into various target spaces.