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Dominating Set on Overlap Graphs of Rectangles Intersecting a Line

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Computing and Combinatorics (COCOON 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11653))

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Abstract

A graph \(G = (V, E)\) is called a rectangle overlap graph if there is a bijection between V and a set \(\mathcal {R}\) of axis-parallel rectangles such that two vertices in V are adjacent if and only if the corresponding rectangles in \(\mathcal {R}\) overlap i.e. their boundaries intersect.

In this article, assuming the Unique Games Conjecture to be true we show that it is not possible to approximate the Minimum Dominating Set (MDS) problem on rectangle overlap graphs with a factor \((2-\epsilon )\) for any \(\epsilon >0\). Previously only APX hardness was known for this problem due to Erlebach and Van Leeuwen (LATIN 2008) and Damian and Pemmaraju (Inf. Process. Lett. 2006). We give an \(O(n^5)\)-time 768-approximation algorithm for the MDS problem on stabbed rectangle overlap graphs i.e. overlap graphs of rectangles intersecting a common straight line. Here n denotes the number of vertices of the input graph.

Our second result is the first constant factor approximation for MDS problem on stabbed rectangle overlap graphs which is a strict generalisation of a graphclass considered by Bandyapadhyay et al. (MFCS 2018).

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Notes

  1. 1.

    Two rectangles R and \(R'\) are non-piercing if both \(R\setminus R'\) and \(R'\setminus R\) are connected.

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Acknowledgement

This research was partially funded by the IFCAM project “Applications of graph homomorphisms” (MA/IFCAM/18/39).

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Correspondence to Dibyayan Chakraborty .

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Chakraborty, D., Das, S., Mukherjee, J. (2019). Dominating Set on Overlap Graphs of Rectangles Intersecting a Line. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_6

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  • DOI: https://doi.org/10.1007/978-3-030-26176-4_6

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