Abstract
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 \(S\subseteq V\) 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 \(S\subseteq V\) 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 \((2\varDelta +1)\)-approximation algorithm for MaxHS on graphs with maximum degree \(\varDelta \). Next, we show that the approximation ratio can be improved to \(\varDelta \) if the input is a connected graph and its maximum degree \(\varDelta \) is a constant. Then, we show that MaxHS can be solved in polynomial time if the input graph is restricted to proper interval graphs, or block graphs. We prove nevertheless that MaxHS remains NP-hard even for bipartite graphs or for cubic graphs.
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