Routing under Balance

Understand

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.