On the Parameterized Complexity of Happy Vertex Coloring - Agrawal, Akanksha (Combinatorial Algorithms, 2018)
Let G be a graph, and $$c: V(G) → [k]$$ be a coloring of vertices in G. A vertex $$u \in V(G)$$ is happy with respect to c if for all $$v \in N_G(u)$$ , we have $$c(u)=c(v)$$ , i.e. all the neighbors of u have color same as that of u. The problem Maximum Happy Vertices takes as an input a graph G, an integer k, a vertex subset $$S \subseteq V(G)$$ , and a (partial) coloring $$c: S → [k]$$ of vertices in S. The goal is to find a coloring $$\tilde{c}: V(G) → [k]$$ such that $$\tilde{c}|_S=c$$ , i.e. $$\tilde{c}$$ extends the partial coloring c to a coloring of vertices in G and the number of happy vertices in G is maximized. For the family of trees, Aravind et al. [1] gave a linear time algorithm for Maximum Happy Vertices for every fixed k, along with the edge variant of the problem. As an open problem, they stated whether Maximum Happy Vertices admits a linear time algorithm on graphs of bounded (constant) treewidth for every fixed k. In this paper, we study the problem Maximum Happy Vertices for graphs of bounded treewidth and give a linear time algorithm for every fixed k and (constant) treewidth of the graph. We also study the problem Maximum Happy Vertices with a different parameterization, which we call Happy Vertex Coloring. The problem Happy Vertex Coloring takes as an input a graph G, integers $$\ell $$ and k, a vertex subset $$S \subseteq V(G)$$ , and a coloring $$c: S → [k]$$ . The goal is to decide if there exist a coloring $$\tilde{c}: V(G) → [k]$$ such that $$\tilde{c}|_S=c$$ and $$|H| \ge \ell $$ , where H is the set of happy vertices in G with respect to $$\tilde{c}$$ . We show that Happy Vertex Coloring is W[1]-hard when parameterized by $$\ell $$ . We also give a kernel for Happy Vertex Coloring with $$\mathcal {O}(k^2\ell ^2)$$ vertices.