The hitting set problem is a generalization of the vertex cover problem to hypergraphs. Xu et al. (Theor Comput Sci 630:117–125, 2016) presented a primal-dual algorithm for the submodular vertex cover problem with linear/submodular penalties. Motivated by their work, we study the submodular hitting set problem with linear penalties (SHSLP). The goal of the SHSLP is to select a vertex subset in the hypergraph to cover some hyperedges and penalize the uncovered ones such that the total cost of covering and penalty is minimized. Based on the primal-dual scheme, we obtain a k-approximation algorithm for the SHSLP, where k is the maximum number of vertices in all hyperedges.

命中集问题是顶点覆盖问题对超图的推广。徐等。（Theor Comput Sci 630：117–125，2016）提出了具有线性/次模罚分的次模顶点覆盖问题的原始-对偶算法。基于他们的工作，我们研究了带有线性罚分（SHSLP）的亚模命中集问题。SHSLP的目标是在超图中选择一个顶点子集以覆盖一些超边并对未覆盖的边进行惩罚，以使覆盖和惩罚的总成本最小化。基于原对偶方案，我们得到了一个ķ为SHSLP，其中近似算法ķ是所有超边的顶点的最大数量。