Abstract

We consider the Minimum Dominating Set (MDS) problem on the intersection graphs of geometric objects. Even for simple and widely-used geometric objects such as rectangles, no sub-logarithmic approximation is known for the problem and (perhaps surprisingly) the problem is NP-hard even when all the rectangles are “anchored” at a diagonal line with slope -1 (Pandit, CCCG 2017). In this paper, we first show that for any ϵ>0, there exists a (2+ϵ)-approximation algorithm for the MDS problem on “diagonal-anchored” rectangles, providing the first O(1)-approximation for the problem on a non-trivial subclass of rectangles. It is not hard to see that the MDS problem on “diagonal-anchored” rectangles is the same as the MDS problem on “diagonal-anchored” L-frames: the union of a vertical and a horizontal line segment that share an endpoint. As such, we also obtain a (2+ϵ)-approximation for the problem with “diagonal-anchored” L-frames.On the other hand, we show that the problem is APX-hard in case the input L-frames intersect the diagonal, or the horizontal segments of the L-frames intersect a vertical line. However, as we show, the problem is linear-time solvable in case the L-frames intersect a vertical as well as a horizontal line. Finally, we consider the MDS problem in the so-called “edge intersection model” and obtain a number of results, answering two questions posed by Mehrabi (WAOA 2017).

Highlights

  • Minimum Dominating Set (MDS) is an NP-hard problem in graph theory and discrete optimization

  • To prove Theorem 4, we show that this class of graphs are the same as permutation graphs for which given the permutation of the vertices, the MDS problem can be solved in linear time [7]

  • In the edge intersection model, the MDS problem on L-frames of a single type is hard to approximate within a factor of 1.1377 even if all the L-frames intersect a vertical line from one side

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Summary

Introduction

Minimum Dominating Set (MDS) is an NP-hard problem in graph theory and discrete optimization. Where two “types” of L-frames are needed for the construction He left open whether the problem remains APX-hard when the input consists of only one type of L-frames or when the L-frames intersect a vertical line. In the edge intersection model, the MDS problem on L-frames of a single type is hard to approximate within a factor of 1.1377 even if all the L-frames intersect a vertical line from one side. We show that even intersecting two lines does not help: the MDS problem is NP-hard on L-frames in the edge intersection model even if every L-frame intersects a vertical line and a horizontal line Observe that this is in contrast to the existence of the linear-time algorithm of Theorem 4 under the standard intersection model. Throughout this paper, the proofs of lemmas and theorems marked with (∗) are given in the full version of the paper due to space constraints

Preliminaries
Diagonal-intersecting Rectangles
Proof of Theorem 1
Proof of Theorem 2
Vertical-intersecting L-frames
A42 A41 A32
Edge Intersection Model
Conclusion
Full Text
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