We consider the Budgeted version of the classical Connected Dominating Set problem (BCDS). Given a graph G and a budget k, we seek a connected subset of at most k vertices maximizing the number of dominated vertices in G. We improve over the previous $(1-e^{-1})/13$ state of the art [Khuller, Purohit, and Sarpatwar, SODA 2014] by introducing a new method for performing tree decompositions in the analysis of the algorithm. This new approach provides a $(1-e^{-1})/12$ approximation guarantee. By generalizing the analysis, we are able to obtain a further improvement to $(1-e^{-7/8})/11$. On the other hand, we prove a $(1-e^{-1}+ϵ)$ inapproximability bound, for any $ϵ>0$, holding also if the subset is required to have a star as a subgraph. In the latter case, we design another algorithm with a matching $(1-e^{-1})$ approximation guarantee.

Also, we examine the edge-vertex domination variant, where an edge dominates its endpoints and all vertices neighboring them. In Budgeted Edge-Vertex Domination (BEVD), we seek a (not necessarily connected) subset of k edges maximizing the number of dominated vertices in G. We prove a $(1-e^{-1})$-approximation and a $(1-e^{-1}+ϵ)$-inapproximability by reductions to/from the maximum coverage problem. Finally, we study the “dual” Partial Edge-Vertex Domination (PEVD) problem, and we present a log-approximation by a reduction to the partial cover problem.

我们考虑经典连通支配集问题（BCDS）的预算版。给定一个图G和一个预算k，我们寻求一个最多k个顶点的连接子集，以最大化G中的主导顶点数量。我们比以前有所改善$（\mathrm{1个}-{Ë}^{-\mathrm{1个}}）/13$最新技术[Khuller，Purohit和Sarpatwar，SODA 2014 ]，方法是在算法分析中引入一种用于执行树分解的新方法。这种新方法提供了$（\mathrm{1个}-{Ë}^{-\mathrm{1个}}）/12$近似保证。通过对分析进行概括，我们可以对$（\mathrm{1个}-{Ë}^{-7/8}）/11$。另一方面，我们证明了$（\mathrm{1个}-{Ë}^{-\mathrm{1个}}+ϵ）$ 对于任何 $ϵ>0$，如果子集需要有一个星星作为子图，则也保持。在后一种情况下，我们设计了另一个具有匹配项的算法$（\mathrm{1个}-{Ë}^{-\mathrm{1个}}）$ 近似保证。

另外，我们检查了边顶点控制的变体，其中边占主导地位的端点和与之相邻的所有顶点。在预算边缘顶点控制（BEVD）中，我们寻求k个边缘的（不一定连接）子集，以最大化G中的主导顶点数量。我们证明$（\mathrm{1个}-{Ë}^{-\mathrm{1个}}）$-近似和 $（\mathrm{1个}-{Ë}^{-\mathrm{1个}}+ϵ）$-通过减少/减少最大覆盖范围问题而无法实现。最后，我们研究了“双重”局部边顶点控制（PEVD）问题，并通过对局部覆盖问题的简化给出了对数逼近。