On disjoint paths in acyclic planar graphs

TL;DR

This paper presents a polynomial-time algorithm for finding disjoint paths in acyclic planar graphs with Eulerian conditions, extending solutions to the integer multiflow problem with fixed number of requests.

Contribution

It introduces a new algorithm with complexity depending on the maximum request and number of edges in the request graph, for acyclic planar digraphs with Eulerian sum of demands and supplies.

Findings

Polynomial algorithm for fixed k when G is acyclic planar and r+c Eulerian.

Complexity depends on maximum request R and number of requests k.

Applicable to arc-disjoint paths problem under specified conditions.

Abstract

We give an algorithm with complexity $O(f(R)^{k^2} k^3 n)$ for the integer multiflow problem on instances $(G,H,r,c)$ with $G$ an acyclic planar digraph and $r+c$ Eulerian. Here, $f$ is a polynomial function, $n = |V(G)|$, $k = |E(H)|$ and $R$ is the maximum request $\max_{h \in E(H)} r(h)$. When $k$ is fixed, this gives a polynomial algorithm for the arc-disjoint paths problem under the same hypothesis.