Interference-free Walks in Time: Temporally Disjoint Paths

TL;DR

This paper studies the complexity of finding multiple temporally disjoint paths in dynamic networks, revealing computational hardness in general cases but also identifying specific tractable scenarios.

Contribution

It provides a detailed complexity analysis of the problem, distinguishing between path and walk versions, and identifies conditions under which the problem is solvable.

Findings

Walk version is W[1]-hard with respect to number of routes

Polynomial-time algorithms exist for a fixed number of walks

Finding two disjoint paths remains NP-hard in general

Abstract

We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never use the same vertex at the same time; otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. On the one extreme, we show that on general graphs the problem is computationally hard. The "walk version" is W[1]-hard when parameterized by the number of routes. However, it is polynomial-time solvable for any constant number of walks. The "path version" remains NP-hard even if we want to find only two temporally disjoint paths. On the other extreme, restricting the input temporal graph to have a path as underlying graph, quite counterintuitively, we find NP-hardness in general but also identify natural tractable cases.