Parameterized Restless Temporal Path

TL;DR

This paper investigates the parameterized complexity of restless temporal paths in temporal graphs, showing NP-hardness in the interval model but providing fixed-parameter tractable algorithms for the point model with various delay conditions.

Contribution

It introduces FPT algorithms for restless temporal paths in the point model and establishes NP-hardness in the interval model, advancing understanding of temporal graph reachability.

Findings

NP-hardness in the interval model for vertex-interval-membership-width of three.

FPT algorithms for the point model with uniform delay one.

FPT algorithms for the point model with arbitrary positive delays.

Abstract

Recently, Bumpus and Meeks introduced a purely temporal parameter, called vertex-interval-membership-width, which is promising for the design of fixed-parameter tractable (FPT) algorithms for vertex reachability problems in temporal graphs. We study this newly introduced parameter for the problem of restless temporal paths, in which the waiting time at each node is restricted. In this article, we prove that, in the interval model, where arcs are present for entire time intervals, finding a restless temporal path is NP-hard even if the vertex-interval-membership-width is equal to three. We exhibit FPT algorithms for the point model, where arcs are present at specific points in time, both with uniform delay one and arbitrary positive delays. In the latter case, this comes with a slight additional computational cost.