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Complexity and Algorithms for ISOMETRIC PATH COVER on Chordal Graphs and Beyond

Authors Dibyayan Chakraborty, Antoine Dailly, Sandip Das, Florent Foucaud , Harmender Gahlawat , Subir Kumar Ghosh

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Shortest paths
  • Isometric path cover
  • Chordal graph
  • Interval graph
  • AT-free graph
  • Approximation algorithm
  • FPT algorithm
  • Treewidth
  • Chordality
  • Treelength

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Abstract

A path is isometric if it is a shortest path between its endpoints. In this article, we consider the graph covering problem Isometric Path Cover, where we want to cover all the vertices of the graph using a minimum-size set of isometric paths. Although this problem has been considered from a structural point of view (in particular, regarding applications to pursuit-evasion games), it is little studied from the algorithmic perspective. We consider Isometric Path Cover on chordal graphs, and show that the problem is NP-hard for this class. On the positive side, for chordal graphs, we design a 4-approximation algorithm and an FPT algorithm for the parameter solution size. The approximation algorithm is based on a reduction to the classic path covering problem on a suitable directed acyclic graph obtained from a breadth first search traversal of the graph. The approximation ratio of our algorithm is 3 for interval graphs and 2 for proper interval graphs. Moreover, we extend the analysis of our approximation algorithm to k-chordal graphs (graphs whose induced cycles have length at most k) by showing that it has an approximation ratio of k+7 for such graphs, and to graphs of treelength at most 𝓁, where the approximation ratio is at most 6𝓁+2.

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Dibyayan Chakraborty, Antoine Dailly, Sandip Das, Florent Foucaud, Harmender Gahlawat, and Subir Kumar Ghosh. Complexity and Algorithms for ISOMETRIC PATH COVER on Chordal Graphs and Beyond. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ISAAC.2022.12

Author Details

Dibyayan Chakraborty
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Antoine Dailly
  • G-SCOP, Univ. Grenoble-Alpes, Grenoble, France
Sandip Das
  • Indian Statistical Institute, Kolkata, India
Florent Foucaud
  • Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
  • Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans Cedex 2, France
Harmender Gahlawat
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
  • Indian Statistical Institute, Kolkata, India
Subir Kumar Ghosh
  • Ramakrishna Mission Vivekananda Educational and Research Institute, Kolkata, India

Funding

This research was financed by the IFCAM project "Applications of graph homomorphisms" (MA/IFCAM/18/39).
  • Foucaud, Florent: This author was financed by the ANR project GRALMECO (ANR-21-CE48-0004-01) and the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25).
  • Gahlawat, Harmender: This author was financed by the ISF grant no. 1176/18.

Acknowledgements

We thank Vincent Limouzy, Joydeep Mukherjee, Lucas Pastor and Jean-Florent Raymond for initial discussions on the topic of this paper.

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