Abstract
We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph $G=(V,E)$ and a collection of $k$ source-destination pairs $\mathcal{M} = \{...
Highlights
Disjoint path problems are well-studied routing problems with several applications and fundamental connections to algorithmic and structural results in combinatorial optimization and graph theory
Canonical problems here are the edge-disjoint paths problem (EDP) and the node-disjoint paths problem (NDP) in undirected graphs. In both these problems the input consists of an undirected graph G = (V, E) and k node-pairs {s1t1, . . . , sktk}
In EDP the goal is to connect the pairs by edge-disjoint paths and in NDP the goal is to connect the pairs by node-disjoint paths
Summary
Disjoint path problems are well-studied routing problems with several applications and fundamental connections to algorithmic and structural results in combinatorial optimization and graph theory. On the theoretical side the model generalizes (modulo constant congestion) the edge and node disjoint paths problems in undirected graphs. The central open question they raised is the following: Is there a poly-logarithmic approximation for Sym-Dir-NDP with constant congestion in general directed graphs? It was shown in [7] that this can be answered in the positive by addressing the following question which is the analogue that was raised in [8] for undirected graphs: If a directed graph G has directed treewidth h, does it have a constant congestion routing structure (crossbar) of size Ω(h/polylog(h))? The main new technical ingredient is a graph theoretic result that shows that if a planar digraph has directed treewidth h it has a constant congestion crossbar of size Ω(h/polylog(h)). We hope that our crossbar result could be used as a starting point to improve the quantitative bound on the grid-minor theorem for planar digraphs
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