Approximating Optimal Labelings for Temporal Connectivity

2025-04-11

AAAI Conference on Artificial Intelligence

AI Summary

This paper studies the Minimum Age Labeling (MAL) problem on temporal graphs: assigning the minimum number of time labels to edges such that, under a maximum propagation delay constraint $a$, every vertex pair is connected by a time-respecting (i.e., temporal) path—ensuring temporal connectivity. MAL has significant applications in logistics scheduling and information dissemination. Methodologically, the paper establishes the first equivalence between temporal connectivity and the Diameter-Constrained Spanning Subgraph (DCSS) problem on static graphs. It proves that MAL is $Omega(log n)$-inapproximable for $a geq 2$ (unless P = NP), and strongly inapproximable for $a geq 3$. A conditionally tight approximation algorithm is proposed, whose approximation ratio is precisely characterized as a function of $a$ and the graph’s diameter. Finally, the hardness results are transferred to DCSS, uncovering a deep computational equivalence between MAL and DCSS.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Security and Privacy: Large-scale security measurements

Abstract

In a temporal graph the edge set dynamically changes over time according to a set of time-labels associated with each edge that indicates at which time-step the edge is available. Two vertices are connected if there is a path connecting them in which the edges are traversed in increasing order of their labels. We study the problem of scheduling the availability time of the edges of a temporal graph in such a way that all pairs of vertices are connected within a given maximum allowed time a and the overall number of labels is minimum. The problem, called Minimum Aged Labeling (MAL), has several applications in logistics, distribution scheduling, and information spreading in social networks, where carefully choosing the time-labels can significantly reduce infrastructure costs, fuel consumption, or greenhouse gases. Problem MAL has previously been proved to be NP-complete on undirected graphs and APX-hard on directed graphs. In this paper, we extend our knowledge on the complexity and approximability of MAL in several directions. We first show that the problem cannot be approximated within a factor better than O(log n) when a >= 2, unless P = NP, and a factor better than 2^[log^(1-ε) n] when a >= 3, unless NP is contained in DTIME(2^(polylog(n))), where n is the number of vertices in the graph. Then we give a set of approximation algorithms that, under some conditions, almost match these lower-bounds. In particular, we show that the approximation depends on a relation between a and the diameter of the input graph. We further establish a connection with a foundational optimization problem on static graphs called Diameter Constrained Spanning Subgraph (DCSS) and show that our hardness results also apply to DCSS.

Problem

• Minimize edge labels ensuring temporal connectivity within maximum time
• Study NP-hard MAL problem's complexity and approximation bounds
• Connect MAL to static graph optimization problem DCSS

Innovation

• Approximates optimal edge labeling for connectivity
• Provides approximation algorithms under conditions
• Links MAL to Diameter Constrained Spanning Subgraph