The Painter's Problem: covering a grid with colored connected polygons (1709.00001v1)

Abstract: Motivated by a new way of visualizing hypergraphs, we study the following problem. Consider a rectangular grid and a set of colors $\chi$. Each cell $s$ in the grid is assigned a subset of colors $\chi_s \subseteq \chi$ and should be partitioned such that for each color $c\in \chi_s$ at least one piece in the cell is identified with $c$. Cells assigned the empty color set remain white. We focus on the case where $\chi = {\text{red},\text{blue}}$. Is it possible to partition each cell in the grid such that the unions of the resulting red and blue pieces form two connected polygons? We analyze the combinatorial properties and derive a necessary and sufficient condition for such a painting. We show that if a painting exists, there exists a painting with bounded complexity per cell. This painting has at most five colored pieces per cell if the grid contains white cells, and at most two colored pieces per cell if it does not.