We study the descending facility location (DFL) problem on the surface of a triangulated terrain. A path from a point s to a point t on the surface of a terrain is descending if the heights of the subsequent points along the path from s to t are in a monotonically non-increasing order [1]. We are given a set $D=\{d_1,d_2,\cdots ,d_n\}$ of n demand points on the surface of a triangulated terrain $\mathcal{W}$ and our objective is to find a set F (of points), of minimum cardinality, on the surface of the terrain such that for each demand point $d\in D$ there exists a descending path from at least one facility $f\in F$ to d. We present an $O((n+m)\log m)$ time algorithm for solving the DFL problem, where m is the number of vertices in the triangulated terrain. We achieve this by reducing the DFL problem to a graph problem called the directed tree covering $(DTC)$ problem. In the DTC problem, we have a directed tree $B=(V,E)$ with a set of marked nodes $M\subseteq V$. The objective is to compute a set $C\subseteq V$ of minimum cardinality, such that for every node $v\in M$, either $v\in C$ or there exists a node $c\in C$ such that v is reachable from c. We prove that the DFL problem can be reduced to DTC problem in $O((m+n)\log m)$ time. The DTC problem thereafter can be solved in $O(|V|)$ time. We also prove that the general version of the DTC problem, called the directed graph covering $(DGC)$ problem is NP-hard on directed bipartite graphs and hard to approximate within $(1-ϵ)\ln |M|$-factor, for every $ϵ>0$, where $|M|$ is the size of the set of marked nodes. We also prove that for the DGC problem, an $O(\log |M|)$ factor approximation is possible and this approximation factor is tight.

我们研究了三角地形表面上的下降设施位置（DFL）问题。如果沿着s到t的路径上后续点的高度按单调非递增顺序[1]，则从地形表面上的点s到点t的路径正在下降。我们得到了一套$d=\{d_{\mathrm{1个}}，d_2，\cdots ，d_{ñ}\}$的ñ三角地形表面上的需求点$\mathcal{w\; ^}$我们的目标是在地形表面上找到最小基数的集合F（由点组成），以便对于每个需求点$d\in d$ 存在至少一个设施的下降路径 $F\in F$到d。我们提出一个$Ø（（ñ+米）\mathrm{日志}米）$解决DFL问题的时间算法，其中m是三角地形中的顶点数。我们通过将DFL问题简化为称为有向树覆盖的图形问题来实现这一目标 $（dŤC）$ 问题。在DTC问题中，我们有一个定向树$乙=（V，Ë）$ 带有一组标记的节点 $\mathrm{中号}\subseteq V$。目的是计算一个集合$C\subseteq V$ 最小基数，例如对于每个节点 $v\in \mathrm{中号}$，或者 $v\in C$ 或存在一个节点 $C\in C$使得v可以从c到达。我们证明DFL问题可以简化为DTC问题$Ø（（米+ñ）\mathrm{日志}米）$时间。此后的DTC问题可以在$Ø（|V|）$时间。我们还证明了DTC问题的一般版本，称为有向图覆盖 $（dGC）$ 问题是NP-在有向二部图上很难实现，并且很难在其中逼近$（\mathrm{1个}-ϵ）\ln |\mathrm{中号}|$因数 $ϵ>0$，在哪里 $|\mathrm{中号}|$是标记节点集的大小。我们还证明，对于DGC问题，$Ø（\mathrm{日志}|\mathrm{中号}|）$ 因子近似是可能的，并且该近似因子很严格。