The Computational Complexity of Finding Temporal Paths under Waiting Time Constraints

TL;DR

This paper studies the computational complexity of finding temporal paths with waiting time constraints in temporal graphs, revealing hardness results and proposing fixed-parameter tractable algorithms for certain parameters.

Contribution

It introduces the concept of $ ext{Δ}$-restless temporal paths, proves their computational hardness, and develops FPT algorithms for various natural parameters, including a new temporal graph parameter.

Findings

Restless temporal path problem is W[1]-hard for certain parameters.

FPT algorithms are provided for maximum path length, feedback edge number, and a new timed feedback vertex number.

The results highlight the complexity landscape and potential tractability in specific parameterized settings.

Abstract

Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration $\Delta$, referred to as $\Delta$-restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the "restless variant" of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.