Applications of Metric Embeddings in Solving Combinatorial Problems

Abstract

Metric embeddings constitute one of the fundamental tools for exploiting the underlying geometric structure of many combinatorial problems. In this dissertation we study some of the applications of metric embeddings in the field of computer science and resolve some of the previously open questions in this area. The results in this dissertation are divided into three parts. In the first part, we study dimension reduction for tree metrics. We show that every $n$-point tree metric admits a $(1+\varepsilon)$ distortion embedding into $\ell_1^{C_\varepsilon \log n}$, for every $\varepsilon > 0$, where $C_\varepsilon = O\left((\frac{1}{\varepsilon})^4 \log \frac{1}{\varepsilon})\right)$. In the case of complete $d$-ary trees we show that this bound can be improved to $C_\varepsilon = O\left(\frac{1}{\varepsilon^2}\right)$. We also show a lower-bound for the dimension required for embedding complete $d$-ary trees into $\ell_1$, which matches the upper bound up to a factor of $O(\log 1/\eps)$. In the second part, we construct two families of metric spaces using the graph product of [Lee and Raghavendra, DCG 2010], and use these constructions to answer two previously open questions. The first construction is used to show that for every $\alpha > 0$ and $n\in \mathbb N$, there exist $n$-point metric spaces $(X,d)$ where every ``scale'' admits a Euclidean embedding with distortion at most $\alpha$, but the whole space requires distortion at least $\Omega(\sqrt{\alpha \log n})$. This shows that the scale-gluing lemma [Lee, SODA 2005] is tight. Previously the matching upper bound was only known for $\alpha=O(1)$ and $\alpha=\Theta(\log n)$. The second construction is used to answer an open problem about negative type metrics. A metric space $(X,d)$ is said to be of negative type if the space $(X,\sqrt{d})$ admits an isometric embedding into $\ell_2$. Metrics of negative type are used to study the power of various inequalities in semi-definite programming relaxations for the Sparsest Cut problem. We exhibit a family of metric spaces $\{(X_m,d_m)\}_{m\in \mathbb N}$ such that $(X_m,\sqrt{d_m})$ admits constant distortion embedding into $\ell_2$, yet it can not be embedded into a metric of negative type with constant distortion. In the last part, we use a new type of random metric embedding to bound the flow and cut gap in node-capacitated planar graphs. The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if all cuts have capacity larger than the demand across the cut. Simple examples show that a similar theorem does not hold if the capacities are on the vertices rather than edges. Nevertheless, we show that there exists a universal constant $\delta > 0$, such that if the equivalent vertex-cut conditions are satisfied, then one can simultaneously route a $\delta$ fraction of flow for all the demands.