Efficient Exact Algorithms for Minimum Covering of Orthogonal Polygons with Squares

TL;DR

A correct reduction with a novel construction of one of the gadgets is provided, and it is shown how this leads to a correct proof of NP-completeness of OPCSH.

Abstract

Let $P$ be an orthogonal polygon of $n$ vertices, without holes. The Orthogonal Polygon Covering with Squares (OPCS) problem takes as input such an orthogonal polygon $P$ with integral vertex coordinates, and asks to find the minimum number of axis-parallel squares whose union is $P$ itself. [Aupperle et. al, 1988] provide an $\mathcal O(N^{1.5})$-time algorithm for OPCS, where $N$ is the number of integral lattice points lying in $P$. In their paper, designing algorithms for OPCS with a running time polynomial in $n$, was stated as an open question; $N$ can be arbitrarily larger than $n$. Output sensitive algorithms were known due to [Bar-Yehuda and Ben-Chanoch, 1994], but these fail to address the open question, as the output can be arbitrarily larger than $n$. We address this open question by designing a polynomial-time exact algorithm for OPCS with a worst-case running time of $\mathcal O(n^{10})$. We also consider the following structural parameterized version of the problem. Let a knob be a polygon edge whose both endpoints are convex polygon vertices. Given an input orthogonal polygon without holes that has $n$ vertices and at most $k$ knobs, we design an algorithm for OPCS with a worst-case running time $\mathcal O(n^2 + k^{10} \cdot n)$. This algorithm is more efficient than the former, whenever $k = o(n^{9/10})$. The problem of Orthogonal Polygon with Holes Covering with Squares (OPCSH) is also studied by [Aupperle et. al, 1988], where the input polygon could have holes. They claim a proof that OPCSH is NP-complete even when the input is the $N$ lattice points inside the polygon. We think there is an error in their proof, where an incorrect reduction from Planar 3-CNF is shown. We provide a correct reduction with a novel construction of one of the gadgets, and show how this leads to a correct proof of NP-completeness of OPCSH.

Figures (9)

• Figure 1: Orthogonal Polygons and Covering with Squares
• Figure 2: Convex vertices: $\{v_1, v_2, v_3, v_5, v_6, v_8, v_{10}, v_{12}, v_{13}\}$, concave vertices: $\{v_4, v_7, v_9, v_{11}, v_{14}\}$
• Figure 3: Orthogonal Polygons and Covering with Squares
• Figure 4: Knobs and Strips
• Figure 5: Rec-packs and Extractions

Theorems & Definitions (65)

• Definition 1: Associated graph $G(P)$
• Proposition 2
• Proposition 3
• Definition 4: Concave and convex vertices
• Definition 5: Valid square
• Definition 6: Maximal Square
• Definition 8: Knob
• Definition 9: Non-knob convex vertex
• Definition 10: Strip
• Definition 12: Rec-pack