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arXiv · 2411.11274

Computing Conforming Partitions with Low Stabbing Number for Rectilinear Polygons

Therese Biedl·Stephane Durocher·Debajyoti Mondal·Rahnuma Islam Nishat·Bastien Rivier
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Abstract

A conforming partition of a rectilinear n-gon P (possibly with holes) is a partition of P into rectangles without using Steiner points (i.e., all corners of all rectangles must lie on the boundary of P). The stabbing number of such a partition is the maximum number of rectangles intersected by an axis-aligned segment lying in the interior of P. In this paper, we examine the problem of computing conforming partitions with low stabbing number. We show that computing a conforming partition with stabbing number at most 4 is NP-hard, which strengthens a previously known hardness result [Durocher \& Mehrabi, Theor. Comput. Sci. 689: 157-168 (2017)] and eliminates the possibility for fixed-parameter-tractable algorithms parameterized by the stabbing number unless P = NP. In contrast, we give (i) an O(n log n)-time algorithm to decide whether a conforming partition with stabbing number 2 exists, (ii) a fixed-parameter-tractable algorithm parameterized by both the stabbing number and treewidth of the pixel graph of the polygon, and (iii) a fixed-parameter-tractable algorithm parameterized by the stabbing number for polygons without holes in general position.

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Therese Biedl, Stephane Durocher, Debajyoti Mondal, Rahnuma Islam Nishat, Bastien Rivier. 2024-11-18. Computing Conforming Partitions with Low Stabbing Number for Rectilinear Polygons. https://arxiv.org/abs/2411.11274

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