The Parameterized Complexity of Happy Colorings

TL;DR

This paper investigates the parameterized complexity of the Maximum Happy Vertex and Edge problems, providing fixed-parameter tractability results for various structural parameters and establishing NP-hardness in specific graph classes.

Contribution

It extends the understanding of the problems' complexity by identifying new fixed-parameter tractable cases based on treewidth, vertex cover, and distance-to-clique parameters.

Findings

FPT algorithms for treewidth and precolored colors

FPT algorithms for vertex cover and distance-to-clique parameters

NP-hardness on split and bipartite graphs, polynomial on cographs

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 maximises 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 \NPH{} on planar graphs and is \FPT{} parameterized by the number of precolored vertices and branchwidth. We continue the study of this problem from a parameterized prespective. Our focus is on both structural and standard parameterizations. To begin with, we establish that the problems are \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 \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 \FPT{} when parameterized by the standard parameter, the number of happy edges. We show that the maximum happy vertex (edge) problem is \NPH{} on split graphs and bipartite graphs and polynomially solvable on cographs.