Routing under Balance

TL;DR

This paper introduces the concept of balance in directed graphs to improve understanding and algorithms for routing and flow problems, providing new algorithms, bounds, and insights into graph structures.

Contribution

It defines the notion of balance in directed graphs and applies it to develop improved oblivious routing schemes and algorithms for maximum flow in balanced graphs.

Findings

Oblivious routing scheme with $O(\alpha \log^3 n / \log \log n)$ competitive ratio for single-source instances.

Lower bounds showing oblivious routing ratios can be as bad as $\Omega(n)$ in general directed graphs.

Fast algorithm for maximum flow in balanced directed graphs.

Abstract

We introduce the notion of balance for directed graphs: a weighted directed graph is $\alpha$-balanced if for every cut $S \subseteq V$, the total weight of edges going from $S$ to $V\setminus S$ is within factor $\alpha$ of the total weight of edges going from $V\setminus S$ to $S$. Several important families of graphs are nearly balanced, in particular, Eulerian graphs (with $\alpha = 1$) and residual graphs of $(1+\epsilon)$-approximate undirected maximum flows (with $\alpha=O(1/\epsilon)$). We use the notion of balance to give a more fine-grained understanding of several well-studied routing questions that are considerably harder in directed graphs. We first revisit oblivious routings in directed graphs. Our main algorithmic result is an oblivious routing scheme for single-source instances that achieve an $O(\alpha \cdot \log^3 n / \log \log n)$ competitive ratio. In the process, we make several technical contributions which may be of independent interest. In particular, we give an efficient algorithm for computing low-radius decompositions of directed graphs parameterized by balance. We also define and construct low-stretch arborescences, a generalization of low-stretch spanning trees to directed graphs. On the negative side, we present new lower bounds for oblivious routing problems on directed graphs. We show that the competitive ratio of oblivious routing algorithms for directed graphs is $\Omega(n)$ in general; this result improves upon the long-standing best known lower bound of $\Omega(\sqrt{n})$ given by Hajiaghayi, Kleinberg, Leighton and R\"acke in 2006. We also show that our restriction to single-source instances is necessary by showing an $\Omega(\sqrt{n})$ lower bound for multiple-source oblivious routing in Eulerian graphs. We also give a fast algorithm for the maximum flow problem in balanced directed graphs.