Approximating Dominating Set on Intersection Graphs of L-frames
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We consider the Dominating Set (DS) problem on the intersection graphs of geometric objects. Surprisingly, for simple and widely used objects such as rectangles, the problem is NP-hard even when all the rectangles are "anchored" at a line with slope -1. It is easy to see that for the anchored rectangles, the problem reduces to one with even simpler objects: L-frames. An L-frame is the union of a vertical and a horizontal segment that share one endpoint (corner of the L-frame). In light of the above discussion, we consider DS on the intersection graphs of L-frames. In this paper, we consider three restricted versions of the problem. First, we consider the version in which the corners of all input L-frames are anchored at a line with slope -1, and obtain a polynomial-time (2+ϵ)-approximation. Furthermore, we obtain a PTAS in case all the input L-frames are anchored at the diagonal from one side. Next, we consider the version, where all input L-frames intersect a vertical line, and prove APX-hardness of this version. Moreover, we prove NP-hardness of this version even in case the horizontal and vertical segments of each L-frame have the same length. Finally, we consider the version, where every L-frame intersects a vertical and a horizontal line, and show that this version is linear-time solvable. We also consider these versions of the problem in the so-called "edge intersection model", and obtain several interesting results. One of the results is an NP-hardness proof of the third version which answers a question posed by Mehrabi (WAOA 2017).
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