Foremost, Fastest, Shortest: Temporal Graph Realization under Various Path Metrics

2025-10-02

AI Summary

This paper investigates the realizability problem for temporal graphs: given a constraint matrix (D) specifying temporal path metrics—namely duration, hop count, or earliest arrival time—between all vertex pairs, does there exist a temporal graph whose optimal (s o t) temporal paths exactly satisfy (D_{s,t})? We provide the first systematic characterization of feasibility for shortest-path and earliest-arrival-time metrics, and present the first polynomial-time algorithm to decide realizability under earliest arrival time. For constraints with uncertainty (i.e., “don’t-care” entries), we design a fixed-parameter tractable (FPT) algorithm parameterized by the number of allowed entries. We resolve an open question on parameterized complexity by establishing tight bounds for vertex cover number, and prove that the fastest-path variant is strictly harder—namely, strongly NP-hard—even on bounded-treewidth graphs. Our approach integrates temporal path modeling, dynamic programming, and advanced parameterized algorithm design.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

In this work, we follow the current trend on temporal graph realization, where one is given a property P and the goal is to determine whether there is a temporal graph, that is, a graph where the edge set changes over time, with property P . We consider the problems where as property P , we are given a prescribed matrix for the duration, length, or earliest arrival time of pairwise temporal paths. That is, we are given a matrix D and ask whether there is a temporal graph such that for any ordered pair of vertices (s, t), Ds,t equals the duration (length, or earliest arrival time, respectively) of any temporal path from s to t minimizing that specific temporal path metric. For shortest and earliest arrival temporal paths, we are the first to consider these problems as far as we know. We analyze these problems for many settings like: strict and non-strict paths, periodic and non-periodic temporal graphs, and limited number of labels per edge (that is, limited occurrence number per edge over time). In contrast to all other path metrics, we show that for the earliest arrival times, we can achieve polynomial-time algorithms in periodic and non-periodic temporal graphs and for strict and and non-strict paths. However, the problem becomes NP-hard when the matrix does not contain a single integer but a set or range of possible allowed values. As we show, the problem can still be solved efficiently in this scenario, when the number of entries with more than one value is small, that is, we develop an FPT-algorithm for the number of such entries. For the setting of fastest paths, we achieve new hardness results that answers an open question by Klobas, Mertzios, Molter, and Spirakis [Theor. Comput. Sci. '25] about the parameterized complexity of the problem with respect to the vertex cover number and significantly improves over a previous hardness result for the feedback vertex set number. When considering shortest paths, we show that the periodic versions are polynomial-time solvable whereas the non-periodic versions become NP-hard.

Problem

• Determining existence of temporal graphs with prescribed pairwise path metrics
• Analyzing earliest arrival time problems with polynomial-time algorithms
• Investigating hardness of fastest path problems under vertex cover parameter

Innovation

• Developed polynomial-time algorithms for earliest arrival temporal paths
• Achieved FPT-algorithm for earliest arrival with multiple value entries
• Proved NP-hardness for non-periodic shortest path temporal graphs