In this paper, we consider the submodular edge cover problem with submodular penalties. In this problem, we are given an undirected graph $G=(V,E)$ with vertex set V and edge set E. Assume the covering cost function $c:2^E\to {\mathbb{R}}_{+}$ and the penalty function $p:2^V\to {\mathbb{R}}_{+}$ are both submodular with p non-decreasing, $c(\varnothing )=0$ and $p(\varnothing )=0$. The goal of the submodular edge cover problem with submodular penalties is to select an edge subset to cover some vertices and penalize the vertex subset containing uncovered vertices such that the total cost of covering and penalty is minimized. For this problem, we first give a 2Δ-approximation algorithm by using a primal-dual technique, where Δ is the maximal degree of the graph G. Then we transform this problem into a submodular set cover problem, and by applying a known result for the submodular set cover problem we conclude that there is an approximation algorithm with an approximation ratio $\mathrm{\Delta }+1$.

在本文中，我们考虑了带有次模块惩罚的次模块边缘覆盖问题。在这个问题上，我们得到一个无向图$G=（\mathrm{伏特}，E）$与顶点组V和边集Ë。承担覆盖成本函数$C：{\mathrm{2个}}^E\to {\mathbb{[R}}_{+}$ 和惩罚函数 $p：{\mathrm{2个}}^{\mathrm{伏特}}\to {\mathbb{[R}}_{+}$都是亚模，且p不递减，$C（\varnothing ）=0$ 和 $p（\varnothing ）=0$。具有次模块惩罚的次模块边缘覆盖问题的目标是选择一个边缘子集来覆盖一些顶点，并对包含未覆盖顶点的顶点子集进行惩罚，以使覆盖和惩罚的总成本最小化。对于这个问题，我们首先使用原始对偶技术给出2Δ近似算法，其中Δ是图G的最大程度。然后，我们将此问题转换为子模集覆盖问题，并通过对子模集覆盖问题应用已知结果，得出结论：存在一种具有近似比率的近似算法$\mathrm{\Delta }+\mathrm{1个}$。