Abstract
We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, i.e., we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In general rectangle stabbing problem, also known as horizontal-vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (i.e., either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan, van Dijk, Fleszar, Spoerhase, and Wolff [5] initiated the study of these problems by providing O(1)-approximation algorithms. Recently, Eisenbrand, Gallato, Svensson, and Venzin [11] have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing but it is open whether the problem admits a PTAS.
In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a \((2+\varepsilon )\)-approximation. We also obtain PTASes for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle’s width is at most its height, and (iii) when all rectangles are \(\delta \)-large, i.e., have at least one edge whose length is at least \(\delta \), while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network.
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Notes
- 1.
The constant is not explicitly stated, and it depends on a not explicitly stated constant in [7].
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Acknowledgement
Arindam Khan was partly supported by Pratiksha Trust Young Investigator Award, Google India Research Award, and Google ExploreCS Award. Andreas Wiese was partially supported by the FONDECYT Regular grant 1200173.
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Khan, A., Subramanian, A., Wiese, A. (2022). A PTAS for the Horizontal Rectangle Stabbing Problem. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_27
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