Theory Seminar

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

Arnold FiltserColumbia University
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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.
In this paper, we propose a novel third bypass called clan embedding. Here each point x is mapped to a subset of points f(x), called a clan, with a special chief point \chi(x)\in f(x). The clan embedding has multiplicative distortion t if for every pair (x,y) some copy y'\in f(y) in the clan of y is  close to the chief of x\min_{y'\in f(y)}d(y',\chi(x))\le t\cdot d(x,y). Our first result is a clan embedding into a tree with multiplicative distortion O(\frac{\log n}{\epsilon}) such that each point has 1+\epsilon 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 D, which were known to be stochastically embeddable into bounded treewidth graphs with expected additive distortion \epsilon D. 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 \rho-dominating set and metric \rho-independent set problems in minor-free graphs.

Joint work with Hung Le

Organizer

Greg Bodwin

Organizer

Euiwoong Lee