Temporal Cycle Detection and Acyclic Temporization

2025-03-04

AI Summary

This paper investigates the definition, detection, and acyclic timestamping of *temporal cycles* in temporal directed graphs. For static directed graphs, it proposes a systematic method to construct timestamp assignments that eliminate all temporal cycles under a bounded lifetime constraint. It formally defines multiple variants of temporal cycles, proves their detection is NP-complete in general (via reductions from 3-SAT and NAE-3-SAT), and designs fixed-parameter tractable (FPT) algorithms. A greedy vertex-ordering–based timestamping strategy is introduced, yielding a constructive theory of acyclic timestamping; an almost-universal polynomial-time algorithm is provided, and its minimal lifetime upper bound is rigorously characterized. The core contributions lie in the tight integration of unified formal modeling, precise computational complexity characterization, and efficient constructive algorithm design.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Detect temporal cycles in directed temporal graphs.
• Time arcs in static digraphs to avoid temporal cycles.
• Bound lifetime in acyclic temporization of temporal digraphs.

Innovation

• Extends cycle concepts to temporal graph theory
• Develops algorithms for temporal cycle detection
• Proposes acyclic temporization with bounded lifetime