  We consider the \emph{Budgeted} version of the classical \emph{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 $(1-1/e)/13$ approximation in [Khuller, Purohit, and Sarpatwar,\ \emph{SODA 2014}] by introducing a new method for performing tree decompositions in the analysis of the last part of the algorithm. This new approach provides a $(1-1/e)/12$ approximation guarantee. By generalizing the analysis of the first part of the algorithm, we are able to modify it appropriately and obtain a further improvement to $(1-e^{-7/8})/11$. On the other hand, we prove a $(1-1/e+\epsilon)$ inapproximability bound, for any $\epsilon > 0$.   We also examine the \emph{edge-vertex domination} variant, where an edge dominates its endpoints and all vertices neighboring them. In \emph{Budgeted Edge-Vertex Domination} (BEVD), we are given a graph $G$, and a budget $k$, and we seek a, not necessarily connected, subset of $k$ edges such that the number of dominated vertices in $G$ is maximized. We prove there exists a $(1-1/e)$-approximation algorithm. Also, for any $\epsilon > 0$, we present a $(1-1/e+\epsilon)$-inapproximability result by a gap-preserving reduction from the \emph{maximum coverage} problem. Finally, we examine the "dual" \emph{Partial Edge-Vertex Domination} (PEVD) problem, where a graph $G$ and a quota $n'$ are given. The goal is to select a minimum-size set of edges to dominate at least $n'$ vertices in $G$. In this case, we present a $H(n')$-approximation algorithm by a reduction to the \emph{partial cover} problem. 
Improved Budgeted Connected Domination and Budgeted Edge-Vertex   Domination