Temporal Cycle Detection and Acyclic Temporization

TL;DR

The paper extends cycle concepts to temporal digraphs by defining multiple temporal-cycle notions and studying two decision problems: Cycle Detection (given a temporal digraph, decide if it contains a temporal cycle) and Acyclic Temporization (time-arranging a static digraph to avoid any temporal cycle, with possible lifetime constraints). It develops polynomial-time and fixed-parameter tractable algorithms, and provides polynomial-time reductions from 3-SAT and Not All Equal 3-SAT to these problems, establishing complexity boundaries. Additionally, the work shows that temporizations derived from arbitrary vertex orderings cover almost all cases, offering practical scheduling heuristics to ensure acyclicity. Overall, the results advance the theory of temporal graphs and yield algorithmic strategies for maintaining acyclic temporal behavior in dynamic networks.

Abstract

In directed graphs, a cycle can be seen as a structure that allows its vertices to loop back to themselves, or as a structure that allows pairs of vertices to reach each other through distinct paths. We extend these concepts to temporal graph theory, resulting in multiple interesting definitions of a "temporal cycle". For each of these, we consider the problems of Cycle Detection and Acyclic Temporization. For the former, we are given an input temporal digraph, and we want to decide whether it contains a temporal cycle. Regarding the latter, for a given input (static) digraph, we want to time the arcs such that no temporal cycle exists in the resulting temporal digraph. We're also interested in Acyclic Temporization where we bound the lifetime of the resulting temporal digraph. Multiple results are presented, including polynomial and fixed-parameter tractable search algorithms, polynomial-time reductions from 3-SAT and Not All Equal 3-SAT, and temporizations resulting from arbitrary vertex orderings which cover (almost) all cases.