Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut
ACM Transactions on Algorithms (TALG)
Coarse Differentiation and Multi-flows in Planar Graphs
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
On the geometry of graphs with a forbidden minor
Proceedings of the forty-first annual ACM symposium on Theory of computing
Flow-cut gaps for integer and fractional multiflows
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Approximating sparsest cut in graphs of bounded treewidth
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
A Tight Upper Bound on the Probabilistic Embedding of Series-Parallel Graphs
SIAM Journal on Discrete Mathematics
Local Versus Global Properties of Metric Spaces
SIAM Journal on Computing
Preserving terminal distances using minors
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Sparsest cut on bounded treewidth graphs: algorithms and hardness results
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Pathwidth, trees, and random embeddings
Combinatorica
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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.