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 $(2\mathrm{\Delta }+1)$-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.

在本文中，我们研究了最大幸福集问题（MaxHS）的逼近度和MaxHS在图类上的计算复杂度：对于无向图$G=（\mathrm{伏特}，E）$ 和一个子集 $\mathrm{小号}\subseteq \mathrm{伏特}$在顶点中，如果v及其所有邻居在S中，则顶点v是快乐的；否则不快乐。给定无向图$G=（\mathrm{伏特}，E）$和一个整数k，MaxHS的目标是找到一个子集$\mathrm{小号}\subseteq \mathrm{伏特}$的ķ顶点这样幸福的顶点数量最大化。MaxHS已知是NP硬性的。在本文中，我们设计了一个$（\mathrm{2个}\mathrm{\Delta }+\mathrm{1个}）$具有最大度数Δ的图上的MaxHS近似估计算法。接下来，我们表明如果输入图的最大程度Δ为常数，则近似比可以提高到Δ。然后，我们表明，如果输入图限制为框图或区间图，则可以在多项式时间内求解MaxHS。但是，我们证明了二部图或立方图上的MaxHS仍然是NP难的。