Minimizing Corners in Colored Rectilinear Grids

Given a rectilinear grid $G$, in which cells are either assigned a single color, out of $k$ possible colors, or remain white, can we color white grid cells of $G$ to minimize the total number of corners of the resulting colored rectilinear polygons in $G$? We show how this problem relates to hypergr...

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Bibliographic Details
Main Authors Depian, Thomas, Dobler, Alexander, Kern, Christoph, Wulms, Jules
Format Journal Article
LanguageEnglish
Published 23.11.2023
Subjects
Computer Science - Computational Geometry
Computer Science - Discrete Mathematics
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Summary:Given a rectilinear grid $G$, in which cells are either assigned a single color, out of $k$ possible colors, or remain white, can we color white grid cells of $G$ to minimize the total number of corners of the resulting colored rectilinear polygons in $G$? We show how this problem relates to hypergraph visualization, prove that it is NP-hard even for $k=2$, and present an exact dynamic programming algorithm. Together with a set of simple kernelization rules, this leads to an FPT-algorithm in the number of colored cells of the input. We additionally provide an XP-algorithm in the solution size, and a polynomial $\mathcal{O}(OPT)$-approximation algorithm.
DOI:10.48550/arxiv.2311.14134