Clan Embeddings into Trees, and Low Treewidth Graphs

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 \emph{clan embedding}. Here each point $x$ is mapped to a subset of points $f(x)$, called a \emph{clan}, with a special \emph{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 prameterized 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.