Disjoint Shortest Paths with Congestion on DAGs

In the k-Disjoint Shortest Paths problem, a set of terminal pairs of vertices $\{(s_i,t_i)\mid 1\le i\le k\}$ is given and we are asked to find paths $P_1,\ldots,P_k$ such that each path $P_i$ is a shortest path from $s_i$ to $t_i$ and every vertex of the graph routes at most one of them. We introduce a generalization of the problem, namely, $k$-Disjoint Shortest Paths with Congestion-$c$ where every vertex is allowed to route up to $c$ paths. We provide a simple algorithm to solve the problem in time $f(k) n^{O(k-c)}$ on DAGs. Using the techniques for DAGs, we show the problem is solvable in time $f(k) n^{O(k)}$ on general undirected graphs. Our algorithm for DAGs is based on the earlier algorithm for $k$-Disjoint Paths with Congestion-$c$[IPL2019], but we significantly simplify their argument. Then we prove that it is not possible to improve the algorithm significantly by showing that for every constant $c$ the problem is W[1]-hard w.r.t.\ parameter $k-c$. We also consider the problem on acyclic planar graphs, but this time we restrict ourselves to the edge-disjoint shortest paths problem. We show that even on acyclic planar graphs there is no $f(k)n^{o(k)}$ algorithm for the problem unless ETH fails.