In Which Graph Structures Can We Efficiently Find Temporally Disjoint Paths and Walks?

TL;DR

This paper investigates the computational complexity of finding temporally disjoint paths and walks in temporal graphs, revealing both hardness results for restricted cases and efficient solutions for simpler scenarios, with implications for multi-agent path finding.

Contribution

It extends previous work by providing parameterized hardness results and identifying cases where the problem can be solved efficiently, highlighting differences between paths and walks.

Findings

Hardness results for restricted cases based on underlying graph parameters

Efficient algorithms for certain simple cases

Differences between path and walk versions of the problem

Abstract

A temporal graph has an edge set that may change over discrete time steps, and a temporal path (or walk) must traverse edges that appear at increasing time steps. Accordingly, two temporal paths (or walks) are temporally disjoint if they do not visit any vertex at the same time. The study of the computational complexity of finding temporally disjoint paths or walks in temporal graphs has recently been initiated by Klobas et al. [IJCAI '21]. This problem is motivated by applications in multi-agent path finding (MAPF), which include robotics, warehouse management, aircraft management, and traffic routing. We extend Klobas et al.'s research by providing parameterized hardness results for very restricted cases, with a focus on structural parameters of the so-called underlying graph. On the positive side, we identify sufficiently simple cases where we can solve the problem efficiently. Our results reveal some surprising differences between the "path version" and the "walk version" (where vertices may be visited multiple times) of the problem, and answer several open questions posed by Klobas et al.