TY - GEN

T1 - Complexity and Algorithms for ISOMETRIC PATH COVER on Chordal Graphs and Beyond

AU - Chakraborty, Dibyayan

AU - Dailly, Antoine

AU - Das, Sandip

AU - Foucaud, Florent

AU - Gahlawat, Harmender

AU - Ghosh, Subir Kumar

N1 - Funding Information:
Funding This research was financed by the IFCAM project “Applications of graph homomorphisms” (MA/IFCAM/18/39). Florent Foucaud: 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). Harmender Gahlawat: This author was financed by the ISF grant no. 1176/18.
Publisher Copyright:
© Dibyayan Chakraborty, Antoine Dailly, Sandip Das, Florent Foucaud, Harmender Gahlawat, and Subir Kumar Ghosh.

PY - 2022/12/1

Y1 - 2022/12/1

N2 - 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.

AB - 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.

KW - Approximation algorithm

KW - AT-free graph

KW - Chordal graph

KW - Chordality

KW - FPT algorithm

KW - Interval graph

KW - Isometric path cover

KW - Shortest paths

KW - Treelength

KW - Treewidth

UR - http://www.scopus.com/inward/record.url?scp=85144209636&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ISAAC.2022.12

DO - 10.4230/LIPIcs.ISAAC.2022.12

M3 - Conference contribution

AN - SCOPUS:85144209636

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 33rd International Symposium on Algorithms and Computation, ISAAC 2022

A2 - Bae, Sang Won

A2 - Park, Heejin

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 33rd International Symposium on Algorithms and Computation, ISAAC 2022

Y2 - 19 December 2022 through 21 December 2022

ER -