Niklas Mohrin

Michelle Döring, Jan Fehse, Tobias Friedrich et al.

The Vehicle Routing Problem (VRP) is a popular generalization of the Traveling Salesperson Problem. Instead of one salesperson traversing the entire weighted, undirected graph $G$, there are $k$ vehicles available to jointly cover the set of clients $C \subseteq V(G)$. Every vehicle must start at one of the depot vertices $D \subseteq V(G)$ and return to its start. Capacitated Vehicle Routing (CVRP) additionally restricts the route of each vehicle by limiting the number of clients it can cover, the distance it can travel, or both. In this work, we study the complexity of VRP and the three variants of CVRP for several parameterizations, in particular focusing on the treewidth of $G$. We present an FPT algorithm for VRP parameterized by treewidth. For CVRP, we prove paraNP- and $W[\cdot]$-hardness for various parameterizations, including treewidth, thereby rendering the existence of FPT algorithms unlikely. In turn, we provide an XP algorithm for CVRP when parameterized by both treewidth and the vehicle capacity.

Michelle Döring, Niklas Mohrin, George Skretas

We introduce the TemporallyEdgeDisjointScheduleCompletion (TEDSC) problem in which we need to cover a set of temporal edge demands $D$ by routing $k$ temporal walks through a directed static graph while remaining temporally edge disjoint. This problem combines the temporal aspects of train routing and passenger demands with the static nature of real-world rail networks. We present a polynomial time algorithm for TEDSC. Motivated by real world constraints, we next investigate two restricted variants of TEDSC in which each walk can only travel for some bounded distance or time $h$. We show that both are tractable when parameterized by $k + h$, but hard for $h$ and $k + |D|$. If we fix the underlying network, the two problems exhibit distinct complexities: The distance variant remains $W[1]$-hard parameterized by $k$ even on a path of three vertices whereas the time variant admits an FPT algorithm on any fixed star. Finally, we show how to approximate the number of required walks up to a factor of $(2-h^{-1})$.