Temporal Routing in Static Networks: The Schedule Completion Problem

2026-04-30

AI Summary

This study addresses the Temporal Edge-Disjoint Path Cover problem (TEDSC) in static directed graphs, which seeks to cover a given set of temporal edge demands \( D \) using \( k \) temporally edge-disjoint temporal paths—an integration of static network topology with temporal scheduling constraints. Through parameterized complexity analysis and algorithm design, the authors establish that TEDSC is solvable in polynomial time. Its variants with distance or time constraints are shown to be fixed-parameter tractable with respect to the combined parameter \( k + h \), yet become intractable when parameterized solely by \( h \) or by \( k + |D| \). For the temporal variant on star networks, an FPT algorithm is provided, along with an approximation algorithm achieving a ratio of \( 2 - 1/h \), thereby revealing fundamental differences in complexity induced by distance versus time constraints.

Abstract

We introduce the TemporallyEdgeDisjointScheduleCompletion (TEDSC) problem in which we need to cover a set of temporal edge demands $D$ by routing $k$ temporal walks through a directed static graph while remaining temporally edge disjoint. This problem combines the temporal aspects of train routing and passenger demands with the static nature of real-world rail networks. We present a polynomial time algorithm for TEDSC. Motivated by real world constraints, we next investigate two restricted variants of TEDSC in which each walk can only travel for some bounded distance or time $h$. We show that both are tractable when parameterized by $k + h$, but hard for $h$ and $k + |D|$. If we fix the underlying network, the two problems exhibit distinct complexities: The distance variant remains $W[1]$-hard parameterized by $k$ even on a path of three vertices whereas the time variant admits an FPT algorithm on any fixed star. Finally, we show how to approximate the number of required walks up to a factor of $(2-h^{-1})$.

Problem

• Temporal Routing
• Schedule Completion
• Temporal Edge Disjoint
• Static Networks
• Temporal Walks

Innovation

• temporal routing
• schedule completion
• parameterized complexity
• temporal edge-disjoint walks
• approximation algorithm