On the approximability of Time Disjoint Walks

We introduce the combinatorial optimization problem Time Disjoint Walks (TDW), which has applications in collision-free routing of discrete objects (e.g., autonomous vehicles) over a network. This problem takes as input a digraph G with positive integer arc lengths, and k pairs of vertices that each represent a trip demand from a source to a destination. The goal is to find a walk and delay for each demand so that no two trips occupy the same vertex at the same time, and so that a min–max or min–sum objective over the trip durations is realized. We focus here on the min–sum variant of Time Disjoint Walks, although most of our results carry over to the min–max case. We restrict our study to various subclasses of DAGs, and observe that there is a sharp complexity boundary between Time Disjoint Walks on oriented stars and on oriented stars with the central vertex replaced by a path. In particular, we present a poly-time algorithm for min–sum and min–max TDW on the former, but show that min–sum TDW on the latter is NP-hard. Our main hardness result is that for DAGs with max degree Δ≤ 3 , min–sum Time Disjoint Walks is APX-hard. We present a natural approximation algorithm for the same class, and provide a tight analysis. In particular, we prove that it achieves an approximation ratio of Θ(k/ log k) on bounded-degree DAGs, and Θ(k) on DAGs and bounded-degree digraphs.