Computing Conforming Partitions with Low Stabbing Number for Rectilinear Polygons

TL;DR

<3-5 sentence high-level summary> The paper investigates the problem of computing conforming partitions of rectilinear polygons (possibly with holes) into rectangles while minimizing the stabbing number of axis-aligned stabbing segments. It establishes NP-hardness for stabbing numbers of at least 4, strengthening prior results, and provides a polynomial-time algorithm for stabbing number 2 plus two fixed-parameter tractable schemes (one parameterized by k+treewidth and one for hole-free general-position polygons). The work also develops a detailed gadget-based hardness reduction and two independent tractability routes for k=2, including a 2-SAT reduction and an O(n log n) algorithm using orthogonal ray-shooting, and studies tractability under bounded treewidth via Courcelle's theorem. It leaves open the case k=3 and the general-hole case for fixed-parameter tractability, offering a roadmap for future exploration in conforming-stabbing problems.>

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.

Figures (25)

• Figure 1: (a) An optimal rectangular partition of a polygon $P _ 1$ in general position with one hole, using two Steiner points (tiny hollow circles) with stabbing number $3$. The portion of the edges of partition rectangles that are not on the boundary of $P _ 1$ are plain bold (red). (b) An optimal conforming partition of $P _ 1$ with stabbing number $4$. (c) The pixel graph of $P _ 1$.
• Figure 2: (a) The reflex segments of $P _ 1$ from Figure \ref{['fig:partitionWithSteinerPointsVSConforming']} are dotted (red) and the reflex vertices are tiny (black) discs. The horizontal and vertical reflex segments $\mathbf{h}_{ p }$ and $\mathbf{v}_{ p }$ from the reflex vertex $p$ are bold. The wedge-pixel of $p$ is shaded (in orange) and labeled $1$. (b) The reflex segments of a thin polygon $P _ 2$ (not in general position) with three holes and seven gates. The gates are the reflex segments with two small orange triangles on their side. (Note that the reflex segment $\overline{ q q ' }$ is not a gate.) (c) An optimal conforming partition of $P _ 2$ with stabbing number $3$.
• Figure 3: An $\mathsf{RPM}\textnormal{-}3\textnormal{-}\mathsf{SAT}$ drawing of $\phi = ( x _ 1 \lor x _ 3 \lor x _ 4 ) \land ( x _ 1 \lor x _ 2 \lor x _ 3 ) \land ( \overline{ x _ 1 } \lor \overline{ x _ 2 } \lor \overline{ x _ 4 } )$.
• Figure 4: The polygon $P ( \phi )$ (not to scale) of the $\mathsf{RPM}\textnormal{-}3\textnormal{-}\mathsf{SAT}$ drawing of $\phi = ( x _ 1 \lor x _ 3 \lor x _ 4 ) \land ( x _ 1 \lor x _ 2 \lor x _ 3 ) \land ( \overline{ x _ 1 } \lor \overline{ x _ 2 } \lor \overline{ x _ 4 } )$. Forcer gadgets are represented by squares labeled F. The reflex segments of a partition with stabbing number $4$ are solid bold (in red), the other reflex segments are dotted (in red). Vertical stabbing segments are bold, and propagate $0$ if they are green (lightly shaded) and $0$ if they are purple (darkly shaded). We use $0 ^ \star$ for a value that is $1$ in the variable assignment but that has been decreased by a variable gadget or by a split gadget and is propagated as $0$.
• Figure 7: The modified (a) variable gadget, (b) split gadget, and (c) clause gadget for polygons in general position and $4\textnormal{-}\mathsf{CSTAB}$.

Theorems & Definitions (33)

• Theorem 3.1
• Lemma 4.1
• proof
• Lemma 4.2: Rules
• proof : Proof of \ref{['R3']}
• proof : Proof of \ref{['R4']}
• Lemma 4.3
• proof
• proof
• Lemma 4.4