There is significant interest in understanding the dynamics of influence diffusion on a social network. The weighted target set selection (WTSS) problem is a fundamental viral marketing problem arising on social networks. In this problem, the goal is to select a set of influential nodes to target (e.g., for promoting a new product) that can influence the rest of the network. The WTSS problem is APX‐hard. With the goal of generating insights to solve the WTSS problem on arbitrary graphs, we study in this paper the WTSS problem on trees and cycles. For trees, we propose a linear‐time dynamic programming algorithm and present a tight and compact extended formulation. Furthermore, we project the extended formulation onto the space of the natural node variables yielding the polytope of the WTSS problem on trees. This projection leads to an exponentially sized set of valid inequalities whose polynomial‐time separation is also discussed. Next, we focus on cycles: we describe a linear‐time algorithm and present the complete description of the polytope for the WTSS problem on cycles. Finally, we describe how these formulations can be applied to arbitrary graphs.

中文翻译：

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树木和周期的加权目标集选择

人们对了解社交网络上影响力扩散的动态非常感兴趣。加权目标集选择（WTSS）问题是社交网络上出现的基本病毒式营销问题。在此问题中，目标是选择一组有影响力的节点作为目标（例如，用于推广新产品），以影响网络的其余部分。WTSS问题是APX难题。为了产生见解以解决任意图上的WTSS问题，我们在本文中研究树和周期上的WTSS问题。对于树木，我们提出了线性时间动态规划算法，并提出了紧密紧凑的扩展公式。此外，我们将扩展公式投影到自然节点变量的空间上，从而在树上产生WTSS问题的多义性。这个投影导致了一组有效不等式的指数大小，并且讨论了多项式与时间的分离。接下来，我们关注循环：我们描述了线性时间算法，并给出了循环中WTSS问题的多面体的完整描述。最后，我们描述如何将这些公式应用于任意图形。