How Many Lines to Paint the City: Exact Edge-Cover in Temporal Graphs

TL;DR

This paper investigates the complexity of covering all edges in a temporal graph with a limited number of journeys, considering various constraints, and provides a comprehensive classification of the problem's computational difficulty.

Contribution

It offers a complete complexity analysis and dichotomies for the temporal edge-cover problem under multiple parameters, advancing understanding of resource minimization in dynamic networks.

Findings

Almost complete complexity classification achieved.

Dichotomies established for various problem variants.

Insights into resource-efficient planning for dynamic networks.

Abstract

Logistics and transportation networks require a large amount of resources to realize necessary connections between locations and minimizing these resources is a vital aspect of planning research. Since such networks have dynamic connections that are only available at specific times, intricate models are needed to portray them accurately. In this paper, we study the problem of minimizing the number of resources needed to realize a dynamic network, using the temporal graphs model. In a temporal graph, edges appear at specific points in time. Given a temporal graph and a natural number k, we ask whether we can cover every temporal edge exactly once using at most k temporal journeys; in a temporal journey consecutive edges have to adhere to the order of time. We conduct a thorough investigation of the complexity of the problem with respect to four dimensions: (a) whether the type of the temporal journey is a walk, a trail, or a path; (b) whether the chronological order of edges in the journey is strict or non-strict; (c) whether the temporal graph is directed or undirected; (d) whether the start and end points of each journey are given or not. We almost completely resolve the complexity of all these problems and provide dichotomies for each one of them with respect to k.