Minimum Temporal Spanners in Happy Graphs

2026-04-27

AI Summary

This study addresses the computational complexity and algorithmic design for the minimum temporal spanner problem in temporal graphs restricted to “happy graphs”—where each edge appears exactly once and every vertex is incident to at most one edge at any time step. Through an NP-hardness reduction, the work establishes for the first time that the problem remains NP-hard even under this restrictive happy graph setting, thereby unifying and strengthening prior complexity results. Additionally, it presents the first positive algorithmic result: when the vertex cover number of the underlying static graph is constant, the problem becomes solvable in polynomial time. Furthermore, the paper proves that in general temporal graphs, parameterizing the problem by the size of a feedback vertex set yields W[1]-hardness, indicating unlikely fixed-parameter tractability under this parameterization.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Topic discovery and tracking

Abstract

Temporal graphs have edge sets that change over discrete time steps. Such graphs are temporally connected (TC) if all pairs of vertices can reach each other using paths that traverse the edges in a time-respecting way (temporal paths). Given a TC temporal graph it, a natural question is to find a minimum spanning subgraph of it that preserves temporal connectivity. These structures, known as temporal spanners, are fundamental and their properties (especially size) have been studied thoroughly in the past decade. In particular, the problem of minimizing the size of a temporal spanner is known to be hard. However, the existing results establish hardness for several incomparable settings and versions of the problem. In this article, we unify and strengthen these results by showing that this problem is NP-hard even on temporal graphs that are simple and proper (also known as "happy"), i.e., where every edge appears only one time, and a vertex cannot be incident to several edges simultaneously. Proving hardness in this extremely restricted setting implies, at once, that the problem is NP-hard for all the previously considered settings and versions of the problem, resolving Open Question 4 in [Casteigts et al. TCS, 2024]. We also initiate the parameterized study of this problem, showing that in the happy setting, the problem can be solved in polynomial time if the underlying graph has a constant-size vertex cover, this result being actually the first positive result on temporal spanners in general. We also show that in the non-happy setting, the problem is W[1]-hard when parameterized by the feedback vertex number of the underlying graph.

Problem

• temporal spanners
• temporal graphs
• NP-hardness
• happy graphs
• temporal connectivity

Innovation

• temporal spanners
• happy graphs
• NP-hardness
• parameterized complexity
• vertex cover