Towards single face shortest vertex-disjoint paths in undirected planar graphs

TL;DR

This paper presents an algorithm for finding shortest vertex-disjoint paths connecting multiple terminal pairs in undirected planar graphs, specifically when all terminals are arranged on a common face, advancing understanding of this complex problem.

Contribution

It provides an $O(kn^5)$ time algorithm for the k disjoint paths problem in planar graphs with terminals on a common face, addressing an open question in the field.

Findings

Algorithm works for terminals on a common face in planar graphs.

Progress towards solving the open problem in planar graph disjoint paths.

Enhances understanding of computational complexity in planar graph routing problems.

Abstract

Given $k$ pairs of terminals $\{(s_{1}, t_{1}), \ldots, (s_{k}, t_{k})\}$ in a graph $G$, the min-sum $k$ vertex-disjoint paths problem is to find a collection $\{Q_{1}, Q_{2}, \ldots, Q_{k}\}$ of vertex-disjoint paths with minimum total length, where $Q_{i}$ is an $s_i$-to-$t_i$ path between $s_i$ and $t_i$. We consider the problem in planar graphs, where little is known about computational tractability, even in restricted cases. Kobayashi and Sommer propose a polynomial-time algorithm for $k \le 3$ in undirected planar graphs assuming all terminals are adjacent to at most two faces. Colin de Verdiere and Schrijver give a polynomial-time algorithm when all the sources are on the boundary of one face and all the sinks are on the boundary of another face and ask about the existence of a polynomial-time algorithm provided all terminals are on a common face. We make progress toward Colin de Verdiere and Schrijver's open question by giving an $O(kn^5)$ time algorithm for undirected planar graphs when $\{(s_{1}, t_{1}), \ldots, (s_{k}, t_{k})\}$ are in counter-clockwise order on a common face.