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Preprints, Working Papers, ... Year : 2022

Complexity and algorithms for Isometric Path Cover on chordal graphs and beyond

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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Dates and versions

hal-03710812 , version 1 (30-06-2022)
hal-03710812 , version 2 (27-10-2022)

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  • HAL Id : hal-03710812 , version 2

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Dibyayan Chakraborty, Antoine Dailly, Sandip Das, Florent Foucaud, Harmender Gahlawat, et al.. Complexity and algorithms for Isometric Path Cover on chordal graphs and beyond. 2022. ⟨hal-03710812v2⟩
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