Approximating Dominating Set on Intersection Graphs of Rectangles and L-frames

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 epsilon>0, there exists a (2+epsilon)-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+epsilon)-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).