The Steiner path problem is a common generalization of the Steiner tree and the Hamiltonian path problem, in which we have to decide if for a given graph there exists a path visiting a fixed set of terminals. In the Steiner cycle problem we look for a cycle visiting all terminals instead of a path. The Steiner path cover problem is an optimization variant of the Steiner path problem generalizing the path cover problem, in which one has to cover all terminals with a minimum number of paths. We study those problems for the special class of interval graphs. We present linear time algorithms for both the Steiner path cover problem and the Steiner cycle problem on interval graphs given as endpoint sorted lists. The main contribution is a lemma showing that backward steps to non-Steiner intervals are never necessary. Furthermore, we show how to integrate this modification to the deferred-query technique of Chang et al. to obtain the linear running times.

施泰纳路径问题是施泰纳树和哈密顿路径问题的常见概括，其中我们必须确定对于给定的图是否存在访问固定终端集的路径。在施泰纳循环问题中，我们寻找访问所有终端的循环而不是路径。Steiner 路径覆盖问题是 Steiner 路径问题的优化变体，它概括了路径覆盖问题，其中必须以最少的路径数覆盖所有终端。我们针对特殊类别的区间图研究这些问题。我们在作为端点排序列表给出的区间图上提出了用于施泰纳路径覆盖问题和施泰纳循环问题的线性时间算法。主要贡献是一个引理，表明非施泰纳区间的后退步骤是不必要的。此外，我们展示了如何将此修改集成到 Chang 等人的延迟查询技术中。获得线性运行时间。