In social networks, the minimum positive influence dominating set (MPIDS) problem is NP-hard, which means it is unlikely to be solved precisely in polynomial time. For the purpose of efficiently solving this problem, greedy approximation algorithms seem appealing because they are fast and can provide guaranteed solutions. In this paper, based on the classic greedy algorithm for cardinality submodular cover, we propose a general greedy approximation algorithm (GGAA) for the MPIDS problem, which uses a generic real-valued submodular potential function, and enjoys a provable approximation guarantee under a wide condition. Two existing greedy algorithms, one of which is unknown for having an approximation ratio, both can be viewed as the specific versions of GGAA, and are shown to enjoy an approximation guarantee of the same order. Applying the framework of GGAA, we also design two new greedy approximation algorithms with fractional submodular potential functions. All these greedy algorithms are \(O(\ln \alpha )\)-approximations where \(\alpha \) is the maximum node degree of the network graph, while it is shown experimentally that these two new algorithms can yield better solutions on typical real social network instances. In this work, as a by-product, we achieve a new approximation ratio of the classic greedy algorithm for cardinality submodular cover, which slightly generalizes two existing results.

在社交网络中，最小正影响支配集 (MPIDS) 问题是 NP-hard 问题，这意味着它不太可能在多项式时间内精确解决。为了有效地解决这个问题，贪婪逼近算法似乎很有吸引力，因为它们速度快并且可以提供有保证的解决方案。在本文中，基于基数子模覆盖的经典贪婪算法，我们针对MPIDS问题提出了一种通用贪婪逼近算法（GGAA），该算法使用通用实值子模势函数，并在广泛的范围内享有可证明的逼近保证健康）状况。现有的两种贪心算法，其中一种是未知的具有逼近比的算法，两者都可以看作是GGAA的特定版本，并被证明享有相同阶数的逼近保证。应用GGAA的框架，我们还设计了两种新的具有分数次模势函数的贪婪逼近算法。所有这些贪心算法都是\(O(\ln \alpha )\) -approximations 其中\(\alpha \)是网络图的最大节点度数，而实验表明这两种新算法可以在典型的真实社交网络实例上产生更好的解决方案. 在这项工作中，作为副产品，我们实现了基数子模覆盖的经典贪婪算法的新近似比，它稍微概括了两个现有结果。