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 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.