In a social network, the propagation of information has sparked intense research. Influence Maximization (IM) is a well-studied problem that asks for k nodes to influence the largest users in the social network. However IM is submodular at the most time. In recent years, many non-submodular problems have been proposed and researchers give a lot of algorithms to solve them. In this paper, we propose Activity Probability Maximization Problem without submodular property. For a given social network G, a candidate edge set \({\overline{E}}\) and a constant k, the Activity Probability Maximization Problem asks for k edges in the candidate edge set that make the all nodes of G with highest probability of being activated under a pre-determined seed set S. Using the marginal increment, we give a general way to construct submodular lower bound and submodular upper bound functions of the non-submodular objective function at the same time. Interestingly, the optimal solution of upper bound is the same as that of lower bound. Therefore, we develop the Sandwich framework called Semi-Sandwich framework. Based on the same optimal solution of lower and upper bounds, we propose a Difference Minimizing Greedy (DMG) algorithm to get an approximation solution of the original problem. Through massive experiments, we show that the method and algorithm are effective.

中文翻译：

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通过社交网络中的建议促进节点活动

在社交网络中，信息的传播引发了广泛的研究。影响力最大化（IM）是一个经过充分研究的问题，要求k个节点影响社交网络中的最大用户。但是，IM最多时间是次模块化的。近年来，已经提出了许多非亚模块问题，研究人员提供了许多算法来解决这些问题。在本文中，我们提出了没有子模性质的活动概率最大化问题。对于给定的社交网络G，候选边缘集\（{\ overline {E}} \）和常数k，活动概率最大化问题要求候选边缘集中的k个边缘构成G的所有节点在预定种子集S下被激活的可能性最高。使用边际增量，我们给出了同时构造非亚模块目标函数的亚模块下界函数和子模块上界函数的一般方法。有趣的是，上限的最优解与下限的最优解相同。因此，我们开发了称为Semi-Sandwich框架的Sandwich框架。基于上下限的相同最优解，我们提出了一种差异最小化贪婪算法（DMG），以获得原始问题的近似解。通过大量实验，表明该方法和算法是有效的。