Parameterized complexity of happy coloring problems

In a vertex-colored graph, an edge is happy if its endpoints have the same color. Similarly, a vertex is happy if all its incident edges are happy. Motivated by the computation of homophily in social networks, we consider the algorithmic aspects of the following MAXIMUM HAPPY EDGES ( k-MHE ) problem: given a partially k-colored graph G and an integer ℓ, find an extended full k-coloring of G making at least ℓ edges happy. When we want to make ℓ vertices happy on the same input, the problem is known as MAXIMUM HAPPY VERTICES ( k-MHV ). We perform an extensive study into the complexity of the problems, particularly from a parameterized viewpoint. For every k≥3, we prove both problems can be solved in time 2ⁿn^O(1). Moreover, by combining this result with a linear vertex kernel of size (k+ℓ) for k-MHE, we show that the edge-variant can be solved in time 2^ℓn^O(1). In contrast, we prove that the vertex-variant remains W[1]-hard for the natural parameter ℓ. However, the problem does admit a kernel with O(k²ℓ²) vertices for the combined parameter k+ℓ. From a structural perspective, we show both problems are fixed-parameter tractable for treewidth and neighborhood diversity, which can both be seen as sparsity and density measures of a graph. Finally, we extend the known [Formula presented]-completeness results of the problems by showing they remain hard on bipartite graphs and split graphs. On the positive side, we show the vertex-variant can be solved optimally in polynomial-time for cographs.