A Polynomial Time Algorithm for the $k$-Disjoint Shortest Paths Problem

The disjoint paths problem is a fundamental problem in algorithmic graph theory and combinatorial optimization. For a given graph $G$ and a set of $k$ pairs of terminals in $G$, it asks for the existence of $k$ vertex-disjoint paths connecting each pair of terminals. The proof of Robertson and Seymour [JCTB 1995] of the existence of an $n^3$ algorithm for any fixed $k$ is one of the highlights of their Graph Minors project. In this paper, we focus on the version of the problem where all the paths are required to be shortest paths. This problem, called the disjoint shortest paths problem, was introduced by Eilam-Tzoreff [DAM 1998] where she proved that the case $k = 2$ admits a polynomial time algorithm. This problem has received some attention lately, especially since the proof of the existence of a polynomial time algorithm in the directed case when $k = 2$ by B\'erczi and Kobayashi [ESA 2017]. However, the existence of a polynomial algorithm when $k = 3$ in the undirected version remained open since 1998. In this paper we show that for any fixed $k$, the disjoint shortest paths problem admits a polynomial time algorithm. In fact for any fixed $C$, the algorithm can be extended to treat the case where each path connecting the pair $(s,t)$ has length at most $d(s,t) + C$.