Vertex Sparsifiers: New Results from Old Techniques

Matthias Englert,Robert Krauthgamer,Harald Räcke,Anupam Gupta,Kunal Talwar,Inbal Talgam-Cohen

doi:10.1137/130908440

Abstract

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 [Proceedings of the 50th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2009, pp. 3--12] and Leighton and Moitra [Proceedings of the 42nd ACM Symposium on Theory of Computing, ACM, New York, 2010, pp. 47--56], we give efficient algorithms for constructing (a) a flow sparsifier $H$ that maintains congestion up to a factor of $O(\frac{\log k}{\log \log k})$, where $k = |K|$; (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$; (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 bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.

Highlights

Given an undirected capacitated graph G = (V, E) and a set of terminal nodes K ⊆ V, we consider the question of producing a graph H only on the terminals K so that the congestion incurred on G and H for any multicommodity flow routed between terminal nodes is similar
We will want the graph H to be structurally “simpler” than G as well. Such a graph H will be called a flow sparsifier for G; the loss of the flow sparsifier is the factor by which the congestions in the graphs G and H differ
We show that any flow sparsifier that only uses edge cap√acities which are bounded from below by a constant, must suffer a loss of Ω( log k/ log log k) in the worst case

Summary

Introduction

We need to give an algorithm that takes as input a graph G = (V, E) with terminals K ⊆ V and outputs a (random) edge-weighted tree T = (K, E) and a retraction f : V → K such that (a ) dT (x, y) ≥ dG(x, y) for all x, y ∈ K (with probability 1), (b ) ET [dT (f (x), f (y))] ≤ O(log k) dG(x, y) for all x, y ∈ V . For every collection of edge capacities c(·) (edge lengths (·)) there exists an efficient algorithm to compute a probabilistic mapping whose congestion (stretch) is, with high probability and in expectation, at most e2ωρ + 1.

Results

Conclusion