Computing Conforming Partitions with Low Stabbing Number for Rectilinear Polygons

TL;DR

This paper investigates the computational complexity of creating rectilinear polygon partitions with low stabbing numbers, proving NP-hardness for stabbing number 4 and providing efficient algorithms for specific cases.

Contribution

It establishes NP-hardness for conforming partitions with stabbing number at most 4 and introduces algorithms for deciding and constructing partitions with low stabbing numbers under certain conditions.

Findings

Computing conforming partitions with stabbing number ≤ 4 is NP-hard.

An O(n log n)-time algorithm decides if a partition with stabbing number 2 exists.

Fixed-parameter algorithms are developed for polygons without holes and for parameters involving stabbing number and treewidth.

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.