Vertex Sparsifiers: New Results from Old Techniques

Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, how should we produce a graph $H$ only on the terminals $K$ so that every (multicommodity) flow between the terminals in $G$ could be supported in $H$ with low congestion, and vice versa? (Such a graph $H$ is called a flow-sparsifier for $G$.) What if we want $H$ to be a "simple" graph? What if we allow $H$ to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier $H$ that maintains congestion up to a factor of $O(\log k/\log \log k)$, where $k = |K|$, (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$, and (c) for a planar graph $G$, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin

Steiner Point Removal with Distortion O(log⁡k)O(\log k)

In the Steiner point removal (SPR) problem, we are given a weighted graph $G=(V,E)$ and a set of terminals $K\subset V$ of size $k$. The objective is to find a minor $M$ of $G$ with only the terminals as its vertex set, such that the distance between the terminals will be preserved up to a small multiplicative distortion. Kamma, Krauthgamer and Nguyen [KKN15] used a ball-growing algorithm with exponential distributions to show that the distortion is at most $O(\log^5 k)$. Cheung [Che17] improved the analysis of the same algorithm, bounding the distortion by $O(\log^2 k)$. We improve the analysis of this ball-growing algorithm even further, bounding the distortion by $O(\log k)$

Separators in Region Intersection Graphs

For undirected graphs G=(V,E) and G_0=(V_0,E_0), say that G is a region intersection graph over G_0 if there is a family of connected subsets {R_u subseteq V_0 : u in V} of G_0 such that {u,v} in E iff R_u cap R_v neq emptyset. We show if G_0 excludes the complete graph K_h as a minor for some h geq 1, then every region intersection graph G over G_0 with m edges has a balanced separator with at most c_h sqrt{m} nodes, where c_h is a constant depending only on h. If G additionally has uniformly bounded vertex degrees, then such a separator is found by spectral partitioning. A string graph is the intersection graph of continuous arcs in the plane. String graphs are precisely region intersection graphs over planar graphs. Thus the preceding result implies that every string graph with m edges has a balanced separator of size O(sqrt{m}). This bound is optimal, as it generalizes the planar separator theorem. It confirms a conjecture of Fox and Pach (2010), and improves over the O(sqrt{m} log m) bound of Matousek (2013)

An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs

Given an edge-weighted graph G with a set Q of k terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the sizes of minimum cuts between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph G being either an arbitrary graph or coming from a specific graph class. In this note we show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with k terminals that require 2^(k-2) edges in any mimicking network. This nearly matches an upper bound of O(k * 2^(2k)) of Krauthgamer and Rika [SODA 2013, arXiv:1702.05951] and is in sharp contrast with the O(k^2) upper bound under the assumption that all terminals lie on a single face [Goranci, Henzinger, Peng, arXiv:1702.01136]. As a side result we show a hard instance for the double-exponential upper bounds given by Hagerup, Katajainen, Nishimura, and Ragde [JCSS 1998], Khan and Raghavendra [IPL 2014], and Chambers and Eppstein [JGAA 2013]

Lower Bounds on 00-Extension with Steiner Nodes

In the $0$-Extension problem, we are given an edge-weighted graph $G=(V,E,c)$, a set $T\subseteq V$ of its vertices called terminals, and a semi-metric $D$ over $T$, and the goal is to find an assignment $f$ of each non-terminal vertex to a terminal, minimizing the sum, over all edges $(u,v)\in E$, the product of the edge weight $c(u,v)$ and the distance $D(f(u),f(v))$ between the terminals that $u,v$ are mapped to. Current best approximation algorithms on $0$-Extension are based on rounding a linear programming relaxation called the \emph{semi-metric LP relaxation}. The integrality gap of this LP, with best upper bound $O(\log |T|/\log\log |T|)$ and best lower bound $\Omega((\log |T|)^{2/3})$, has been shown to be closely related to the best quality of cut and flow vertex sparsifiers. We study a variant of the $0$-Extension problem where Steiner vertices are allowed. Specifically, we focus on the integrality gap of the same semi-metric LP relaxation to this new problem. Following from previous work, this new integrality gap turns out to be closely related to the quality achievable by cut/flow vertex sparsifiers with Steiner nodes, a major open problem in graph compression. Our main result is that the new integrality gap stays superconstant $\Omega(\log\log |T|)$ even if we allow a super-linear $O(|T|\log^{1-\varepsilon}|T|)$ number of Steiner nodes