Abstract
Graph sparsification aims at compressing large graphs into smaller ones while preserving important characteristics of the input graph. In this work we study vertex sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. We focus on the following notions: (1) Given a digraph $G=(V,E)$ and terminal vertices $K \subset V$ with $|K| = k$, a (vertex) reachability sparsifier of $G$ is a digraph $H=(V_H,E_H)$, $K \subset V_H$ that preserves all reachability information among terminal pairs. Let $|V_H|$ denote the size of $H$. In this work we introduce the notion of reachability-preserving minors (RPMs), i.e., we require $H$ to be a minor of $G$. We show any directed graph $G$ admits an RPM $H$ of size $O(k^3)$, and if $G$ is planar, then the size of $H$ improves to $O(k^{2} \log k)$. We complement our upper bound by showing that there exists an infinite family of grids such that any RPM must have $\Omega(k^{2})$ vertices. (2) Given a weighted undirected graph $G=(V,E)$ and terminal vertices $K$ with $|K|=k$, an exact (vertex) cut sparsifier of $G$ is a graph $H$ with $K \subset V_H$ that preserves the value of minimum cuts separating any bipartition of $K$. We show that planar graphs with all the $k$ terminals lying on the same face admit exact cut sparsifiers of size $O(k^{2})$ that are also planar. Our result extends to flow and distance sparsifiers. It improves the previous best-known bound of $O(k^22^{2k})$ for cut and flow sparsifiers by an exponential factor and matches an $\Omega(k^2)$ lower-bound for this class of graphs.
Highlights
Very large graphs or networks are ubiquitous nowadays, from social networks to information networks
We introduce the notion of reachability-preserving minors, i.e., we require H to be a minor of G
Typical examples include cut sparsifiers [3], spectral sparsifiers [36], spanners [40] and transitive reductions [1], which are subgraphs defined on the same vertex set of the original graph G while having much smaller number of edges and still well preserving the cut structure, spectral properties, pairwise distances, transitive closure of G, respectively
Summary
Very large graphs or networks are ubiquitous nowadays, from social networks to information networks. For any OS instance G, i.e., a k-terminal planar graph in which all terminals lie on the same face, there exist quality-1 vertex cut, flow and distance sparsifers of size O(k2). And independently of our work, Krauthgamer and Rika [28] constructed quality-1 cut sparsifiers of size O(γ522γk4) for planar graphs whose terminals are incident to at most γ = γ(G) faces. For the quality 1 distance sparsifiers, Krauthgamer, Nguyen and Zondiner [26] introduced the notion of distance-preserving minors, and showed an upperbound of size O(k4) for general undirected graphs They gave a lower bound of Ω(k2) on the size of such a minor for planar graphs. Over the last two decades, there has been a considerable amount of work on understanding the tradeoff between the sparsifier’s quality q and its size for q > 1, i.e., when the sparsifiers only approximately preserve the corresponding properties [11, 2, 34, 31, 6, 17, 32, 21, 7, 5, 17, 23, 10, 16]
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