#### Theory Seminar

# Arnold Filtser: Clan Embeddings into Trees, and Low Treewidth Graphs

Arnold FiltserColumbia University

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November 3, 2021 @ 11:00 am - 12:00 pm

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In low distortion metric embeddings, the goal is to embed a host “hard” metric space into a “simpler” target space while approximately preserving pairwise distances. A highly desirable target space is that of a tree metric. Unfortunately, such embedding will result in a huge distortion.

A celebrated bypass to this problem is stochastic embedding with logarithmic expected distortion. Another bypass is Ramsey-type embedding, where the distortion guarantee applies only to a subset of the points. However, both these solutions fail to provide an embedding into a single tree with a worst-case distortion guarantee on all pairs.

A celebrated bypass to this problem is stochastic embedding with logarithmic expected distortion. Another bypass is Ramsey-type embedding, where the distortion guarantee applies only to a subset of the points. However, both these solutions fail to provide an embedding into a single tree with a worst-case distortion guarantee on all pairs.

In this paper, we propose a novel third bypass called

*clan embedding*. Here each point is mapped to a subset of points , called a*clan*, with a special chief point . The clan embedding has multiplicative distortion if for every pair some copy in the clan of is close to the chief of : . Our first result is a clan embedding into a tree with multiplicative distortion such that each point has copies (in expectation). In addition, we provide a “spanning” version of this theorem for graphs and use it to devise the first compact routing scheme with constant size routing tables.We then focus on minor-free graphs of diameter parameterized by , which were known to be stochastically embeddable into bounded treewidth graphs with expected additive distortion . We devise Ramsey-type embedding and clan embedding analogs of the stochastic embedding. We use these embeddings to construct the first (bicriteria quasi-polynomial time) approximation scheme for the metric -dominating set and metric -independent set problems in minor-free graphs.

Joint work with Hung Le