The Parameterized Complexity of Happy Colorings

Abstract

Consider a graph $G = (V,E)$ and a coloring c of vertices with colors from $[\ell ]$. A vertex v is said to be happy with respect to c if $c(v) = c(u)$ for all neighbors u of v. Further, an edge (u, v) is happy if $c(u) = c(v)$. Given a partial coloring c of V, the Maximum Happy Vertex (Edge) problem asks for a total coloring of V extending c to all vertices of V that maximizes the number of happy vertices (edges). Both problems are known to be NP-hard in general even when $\ell = 3$, and is polynomially solvable when $\ell = 2$. In [IWOCA 2016] it was shown that both problems are polynomially solvable on trees, and for arbitrary k, it was shown that MHE is NP-hard on planar graphs and is $\mathsf {FPT}$ parameterized by the number of precolored vertices and branchwidth.

We continue the study of this problem from a parameterized perspective. Our focus is on both structural and standard parameterizations. To begin with, we establish that the problems are $\mathsf {FPT}$ when parameterized by the treewidth and the number of colors used in the precoloring, which is a potential improvement over the total number of precolored vertices. Further, we show that both the vertex and edge variants of the problem is $\mathsf {FPT}$ when parameterized by vertex cover and distance-to-clique parameters. We also show that the problem of maximizing the number of happy edges is $\mathsf {FPT}$ when parameterized by the standard parameter, the number of happy edges. We show that the maximum happy vertex (edge) problem is NP-hard on split graphs and bipartite graphs and maximum happy vertex problem is polynomially solvable on cographs.