The Painter's Problem: covering a grid with colored connected polygons

TL;DR

This paper investigates a grid coloring problem inspired by hypergraph visualization, establishing conditions for partitioning cells into connected colored polygons with bounded complexity.

Contribution

It provides a necessary and sufficient condition for the existence of such a coloring and bounds the number of pieces per cell.

Findings

Existence of a coloring depends on a specific combinatorial condition.

If a coloring exists, it can be realized with at most five pieces per cell with white cells.

In absence of white cells, at most two pieces per cell are sufficient.

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.