We consider a variant of the prize collecting Steiner tree problem in which we are given a \emph{directed graph} $D=(V,A)$, a monotone submodular prize function $p:2^V \rightarrow \mathbb{R}^+ \cup \{0\}$, a cost function $c:V \rightarrow \mathbb{Z}^{+}$, a root vertex $r \in V$, and a budget $B$. The aim is to find an out-subtree $T$ of $D$ rooted at $r$ that costs at most $B$ and maximizes the prize function. We call this problem \emph{Directed Rooted Submodular Tree} (\textbf{DRSO}). Very recently, Ghuge and Nagarajan [SODA\ 2020] gave an optimal quasi-polynomial-time $O\left(\frac{\log n'}{\log \log n'}\right)$-approximation algorithm, where $n'$ is the number of vertices in an optimal solution, for the case in which the costs are associated to the edges. In this paper, we give a polynomial-time algorithm for \textbf{DRSO} that guarantees an approximation factor of $O(\sqrt{B}/\epsilon^3)$ at the cost of a budget violation of a factor $1+\epsilon$, for any $\epsilon \in (0,1]$. The same result holds for the edge-cost case, to the best of our knowledge this is the first polynomial-time approximation algorithm for this case. We further show that the unrooted version of \textbf{DRSO} can be approximated to a factor of $O(\sqrt{B})$ without budget violation, which is an improvement over the factor $O(\Delta \sqrt{B})$ given in~[Kuo et al.\ IEEE/ACM\ Trans.\ Netw.\ 2015] for the undirected and unrooted case, where $\Delta$ is the maximum degree of the graph. Finally, we provide some new/improved approximation bounds for several related problems, including the additive-prize version of \textbf{DRSO}, the maximum budgeted connected set cover problem, and the budgeted sensor cover problem.

翻译：我们考虑一个收集 Steiner 树质问题的奖项变量, 其中我们得到的是 $D = (V, A)$, 一个单调的子模式奖项函数 $p: 2V\ right \ mathb{R\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\美元, 一个成本函数 : V\ right $\\\ 美元, 一个根顶替的顶替性(\\\\\\\\\\\\\\\\\ 方向) 美元, 基底值$美元 $($B), 基值的美元(美元) 基值值 。