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Complexity of the Temporal Shortest Path Interdiction Problem

Authors Jan Boeckmann , Clemens Thielen , Alina Wittmann

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Jan Boeckmann
  • TUM Campus Straubing for Biotechnology and Sustainability, Hochschule Weihenstephan-Triesdorf, Germany
Clemens Thielen
  • TUM Campus Straubing for Biotechnology and Sustainability, Hochschule Weihenstephan-Triesdorf, Germany
  • Department of Mathematics, School of Computation, Information and Technology, Technische Universität München, Germany
Alina Wittmann
  • Department of Mathematics, School of Computation, Information and Technology, Technische Universität München, Germany

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Jan Boeckmann, Clemens Thielen, and Alina Wittmann. Complexity of the Temporal Shortest Path Interdiction Problem. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SAND.2023.9

Abstract

In the shortest path interdiction problem, an interdictor aims to remove arcs of total cost at most a given budget from a directed graph with given arc costs and traversal times such that the length of a shortest s-t-path is maximized. For static graphs, this problem is known to be strongly NP-hard, and it has received considerable attention in the literature. While the shortest path problem is one of the most fundamental and well-studied problems also for temporal graphs, the shortest path interdiction problem has not yet been formally studied on temporal graphs, where common definitions of a "shortest path" include: latest start path (path with maximum start time), earliest arrival path (path with minimum arrival time), shortest duration path (path with minimum traveling time including waiting times at nodes), and shortest traversal path (path with minimum traveling time not including waiting times at nodes). In this paper, we analyze the complexity of the shortest path interdiction problem on temporal graphs with respect to all four definitions of a shortest path mentioned above. Even though the shortest path interdiction problem on static graphs is known to be strongly NP-hard, we show that the latest start and the earliest arrival path interdiction problems on temporal graphs are polynomial-time solvable. For the shortest duration and shortest traversal path interdiction problems, however, we show strong NP-hardness, but we obtain polynomial-time algorithms for these problems on extension-parallel temporal graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Theory of computation → Dynamic graph algorithms
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Temporal Graphs
  • Interdiction Problems
  • Complexity
  • Shortest Paths
  • Most Vital Arcs

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