Path cover is a well-known intractable problem that finds a minimum number of vertex disjoint paths in a given graph to cover all the vertices. We show that a variant, in which the objective is to minimize the number of length-0 paths, is polynomial-time solvable. We further show that another variant, to minimize the total number of length-0 and length-1 paths, is also polynomial-time solvable. Both variants find applications in approximating the two-machine flow-shop scheduling problem in which job processing has constraints that are formulated as a conflict graph. For the unit jobs, we present a 4/3-approximation for the scheduling problem with an arbitrary conflict graph, based on the exact algorithm for the above second variant of the path cover problem. For arbitrary jobs where the conflict graph is the union of two disjoint cliques, we present a simple 3/2-approximation algorithm.

路径覆盖是一个众所周知的棘手问题，它在给定的图中找到最少数量的顶点不相交路径来覆盖所有顶点。我们展示了一个变体，其目标是最小化长度为 0 的路径的数量，是多项式时间可解的。我们进一步展示了另一个变体，以最小化长度为 0 和长度为 1 的路径的总数，也是多项式时间可解的。两种变体都可以应用于近似两机流水车间调度问题，其中作业处理具有被表述为冲突图的约束。对于单元作业，我们基于路径覆盖问题的上述第二个变体的精确算法，提出了具有任意冲突图的调度问题的 4/3 近似。对于冲突图是两个不相交集团的联合的任意工作，