101 research outputs found
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]
Improved guarantees for Vertex Sparsification in planar graphs
Graph Sparsification aims at compressing large graphs into smaller ones while (approximately) 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. Given a weighted graph G=(V,E), and a terminal set K with |K|=k, a quality-q vertex cut sparsifier of G is a graph H with K contained in V_H that preserves the value of minimum cuts separating any bipartition of K, up to a factor of q. We show that planar graphs with all the k terminals lying on the same face admit quality-1 vertex cut sparsifier of size O(k^2) that are also planar. Our result extends to vertex flow and distance sparsifiers. It improves the previous best known bound of O(k^2 2^(2k)) for cut and flow sparsifiers by an exponential factor, and matches an Omega(k^2) lower-bound for this class of graphs. We also study vertex reachability sparsifiers for directed graphs. Given a digraph G=(V,E) and a terminal set K, a vertex reachability sparsifier of G is a digraph H=(V_H,E_H), K contained in V_H that preserves all reachability information among terminal pairs. We introduce the notion of reachability-preserving minors, i.e., we require H to be a minor of G. Among others, for general planar digraphs, we construct reachability-preserving minors of size O(k^2 log^2 k). We complement our upper-bound by showing that there exists an infinite family of acyclic planar digraphs such that any reachability-preserving minor must have Omega(k^2) vertices
Steiner Point Removal with Distortion
In the Steiner point removal (SPR) problem, we are given a weighted graph
and a set of terminals of size . The objective is to
find a minor of 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 . Cheung [Che17] improved the analysis of
the same algorithm, bounding the distortion by . We improve the
analysis of this ball-growing algorithm even further, bounding the distortion
by
Degree-3 Treewidth Sparsifiers
We study treewidth sparsifiers. Informally, given a graph of treewidth
, a treewidth sparsifier is a minor of , whose treewidth is close to
, is small, and the maximum vertex degree in is bounded.
Treewidth sparsifiers of degree are of particular interest, as routing on
node-disjoint paths, and computing minors seems easier in sub-cubic graphs than
in general graphs.
In this paper we describe an algorithm that, given a graph of treewidth
, computes a topological minor of such that (i) the treewidth of
is ; (ii) ; and (iii) the maximum
vertex degree in is . The running time of the algorithm is polynomial in
and . Our result is in contrast to the known fact that unless , treewidth does not admit polynomial-size kernels.
One of our key technical tools, which is of independent interest, is a
construction of a small minor that preserves node-disjoint routability between
two pairs of vertex subsets. This is closely related to the open question of
computing small good-quality vertex-cut sparsifiers that are also minors of the
original graph.Comment: Extended abstract to appear in Proceedings of ACM-SIAM SODA 201
Mimicking Networks and Succinct Representations of Terminal Cuts
Given a large edge-weighted network with terminal vertices, we wish
to compress it and store, using little memory, the value of the minimum cut (or
equivalently, maximum flow) between every bipartition of terminals. One
appealing methodology to implement a compression of is to construct a
\emph{mimicking network}: a small network with the same terminals, in
which the minimum cut value between every bipartition of terminals is the same
as in . This notion was introduced by Hagerup, Katajainen, Nishimura, and
Ragde [JCSS '98], who proved that such of size at most always
exists. Obviously, by having access to the smaller network , certain
computations involving cuts can be carried out much more efficiently.
We provide several new bounds, which together narrow the previously known gap
from doubly-exponential to only singly-exponential, both for planar and for
general graphs. Our first and main result is that every -terminal planar
network admits a mimicking network of size , which is
moreover a minor of . On the other hand, some planar networks require
. For general networks, we show that certain bipartite
graphs only admit mimicking networks of size , and
moreover, every data structure that stores the minimum cut value between all
bipartitions of the terminals must use machine words
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