The growing importance of online social networks where people share information with others leads to the emergence of viral marketing, a new way to promote the sales of products. A derivation of classical Influence Maximization (IM) problem is the Profit Maximization (PM) problem that we focus on in this paper. We propose the PM problem with a cardinality constraint in order to make it closer to the real marketing activities. Without a fixed and pre-determined budget for seed selection, the profit spread metric of PM considers the total benefit and cost. The difference between influence spread metric and profit spread metric is that the latter is no longer monotone and lose the property of submodularity in general. Due to the natural form as the difference between two submodular functions, the profit spread metric admits a DS decomposition. What matters is that we design a Marginal increment-based Prune and Search (MPS) algorithm. From the perspective of marginal increment, MPS algorithm can compute profit spread more directly and accurately. Extensive experiments demonstrate the effectiveness and outperformance of our algorithm.

人们与他人共享信息的在线社交网络的重要性日益增加，这导致了病毒式营销的出现，这是一种促进产品销售的新方式。经典影响最大化 (IM) 问题的衍生是我们在本文中关注的利润最大化 (PM) 问题。我们提出了具有基数约束的 PM 问题，以使其更接近真实的营销活动。在没有固定和预先确定的种子选择预算的情况下，PM 的利润分布指标考虑了总收益和成本。影响力扩散度量和利润扩散度量的区别在于后者不再是单调的，并且总体上失去了子模块性的特性。由于作为两个子模函数之间差异的自然形式，利润传播度量允许 DS 分解。重要的是我们设计了一个基于边际增量的修剪和搜索 (MPS) 算法。从边际增量的角度来看，MPS 算法可以更直接、更准确地计算利润点差。大量实验证明了我们算法的有效性和优异性能。