In this article we consider the Directed Steiner Path Cover problem on directed co-graphs. Given a directed graph \(G=(V,E)\) and a set \(T \subseteq V\) of so-called terminal vertices, the problem is to find a minimum number of vertex-disjoint simple directed paths, which contain all terminal vertices and a minimum number of non-terminal vertices (Steiner vertices). The primary minimization criteria is the number of paths. We show how to compute in linear time a minimum Steiner path cover for directed co-graphs. This leads to a linear time computation of an optimal directed Steiner path on directed co-graphs, if it exists. Since the Steiner path problem generalizes the Hamiltonian path problem, our results imply the first linear time algorithm for the directed Hamiltonian path problem on directed co-graphs. We also give binary integer programs for the (directed) Hamiltonian path problem, for the (directed) Steiner path problem, and for the (directed) Steiner path cover problem. These integer programs can be used to minimize change-over times in pick-and-place machines used by companies in electronic industry.

在本文中，我们考虑有向共图上的有向 Steiner 路径覆盖问题。给定一个有向图\(G=(V,E)\)和一个集合\(T \subseteq V\)对于所谓的终端顶点，问题是找到最少数量的顶点不相交的简单有向路径，其中包含所有终端顶点和最少数量的非终端顶点（Steiner 顶点）。主要的最小化标准是路径数。我们展示了如何在线性时间内计算有向共图的最小 Steiner 路径覆盖。如果存在，这将导致对有向共图上的最佳有向 Steiner 路径进行线性时间计算。由于 Steiner 路径问题推广了哈密顿路径问题，我们的结果暗示了有向共图上有向哈密顿路径问题的第一个线性时间算法。我们还给出了（有向）哈密顿路径问题、（有向）Steiner 路径问题和（有向）Steiner 路径覆盖问题的二进制整数程序。