# On the Approximability of Time Disjoint Walks

We introduce the combinatorial optimization problem Time Disjoint Walks. 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 path and delay for each demand so that no two trips occupy the same vertex at the same time, and so that the sum of trip times is minimized. We show that even for DAGs with max degree Δ≤3, Time Disjoint Walks is APX-hard. We also present a natural approximation algorithm, and provide a tight analysis. In particular, we prove that it achieves an approximation ratio of Θ(k/ k) on bounded-degree DAGs, and Θ(k) on DAGs and bounded-degree digraphs.

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