Complexity and approximability of the happy set problem.

• The approximability and the computational complexity of the Maximum Happy Set problem (MaxHS) are studied. • A (2 Δ + 1) -approximation algorithm for MaxHS on graphs with maximum degree Δ is presented. • The approximation ratio is improved to Δ if the maximum degree Δ is a constant. • The polynomial-time solvability of MaxHS on block/interval graphs is proved. • The NP-hardness of MaxHS on bipartite/cubic graphs is proved. In this paper we study the approximability of the Maximum Happy Set problem (MaxHS) and the computational complexity of MaxHS on graph classes: For an undirected graph G = (V , E) and a subset S ⊆ V of vertices, a vertex v is happy if v and all its neighbors are in S ; otherwise unhappy. Given an undirected graph G = (V , E) and an integer k , the goal of MaxHS is to find a subset S ⊆ V of k vertices such that the number of happy vertices is maximized. MaxHS is known to be NP-hard. In this paper, we design a (2 Δ + 1) -approximation algorithm for MaxHS on graphs with maximum degree Δ. Next, we show that the approximation ratio can be improved to Δ if the maximum degree Δ of the input graph is a constant. Then, we show that MaxHS can be solved in polynomial time if the input graph is restricted to block graphs, or interval graphs. We prove nevertheless that MaxHS on bipartite graphs or on cubic graphs remains NP-hard. [ABSTRACT FROM AUTHOR]