Metric Embedding via Shortest Path Decompositions

We study the problem of embedding shortest-path metrics of weighted graphs into $\ell_p$ spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of Shortest Path Decomposition depth is inductively defined: A (weighed) path graph has shortest path decomposition (SPD) depth $1$. General graph has an SPD of depth $k$ if it contains a shortest path whose deletion leads to a graph, each of whose components has SPD depth at most $k-1$. In this paper we give an $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$-distortion embedding for graphs of SPD depth at most $k$. This result is asymptotically tight for any fixed $p>1$, while for $p=1$ it is tight up to second order terms. As a corollary of this result, we show that graphs having pathwidth $k$ embed into $\ell_p$ with distortion $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$. For $p=1$, this improves over the best previous bound of Lee and Sidiropoulos that was exponential in $k$; moreover, for other values of $p$ it gives the first embeddings whose distortion is independent of the graph size $n$. Furthermore, we use the fact that planar graphs have SPD depth $O(\log n)$ to give a new proof that any planar graph embeds into $\ell_1$ with distortion $O(\sqrt{\log n})$. Our approach also gives new results for graphs with bounded treewidth, and for graphs excluding a fixed minor

One Tree to Rule Them All: Poly-Logarithmic Universal Steiner Tree

A spanning tree $T$ of graph $G$ is a $\rho$-approximate universal Steiner tree (UST) for root vertex $r$ if, for any subset of vertices $S$ containing $r$, the cost of the minimal subgraph of $T$ connecting $S$ is within a $\rho$ factor of the minimum cost tree connecting $S$ in $G$. Busch et al. (FOCS 2012) showed that every graph admits $2^{O(\sqrt{\log n})}$-approximate USTs by showing that USTs are equivalent to strong sparse partition hierarchies (up to poly-logs). Further, they posed poly-logarithmic USTs and strong sparse partition hierarchies as open questions. We settle these open questions by giving polynomial-time algorithms for computing both $O(\log ^ 7 n)$-approximate USTs and poly-logarithmic strong sparse partition hierarchies. For graphs with constant doubling dimension or constant pathwidth we improve this to $O(\log n)$-approximate USTs and $O(1)$ strong sparse partition hierarchies. Our doubling dimension result is tight up to second order terms. We reduce the existence of these objects to the previously studied cluster aggregation problem and what we call dangling nets.Comment: @FOCS2

Non-Homogenizable Classes of Finite Structures

Homogenization is a powerful way of taming a class of finite structures with several interesting applications in different areas, from Ramsey theory in combinatorics to constraint satisfaction problems (CSPs) in computer science, through (finite) model theory. A few sufficient conditions for a class of finite structures to allow homogenization are known, and here we provide a necessary condition. This lets us show that certain natural classes are not homogenizable: 1) the class of locally consistent systems of linear equations over the two-element field or any finite Abelian group, and 2) the class of finite structures that forbid homomorphisms from a specific MSO-definable class of structures of treewidth two. In combination with known results, the first example shows that, up to pp-interpretability, the CSPs that are solvable by local consistency methods are distinguished from the rest by the fact that their classes of locally consistent instances are homogenizable. The second example shows that, for MSO-definable classes of forbidden patterns, treewidth one versus two is the dividing line to homogenizability