Inspired by the famous Target Set Selection problem, we propose a new discrete model to simultaneously spread two opinions within a social network and perform an initial study of its complexity. Here, we are given a social network, a seed-set of agents for each opinion, two thresholds for each agent, a budget, and a number of rounds. The first threshold represents the willingness of an agent to adopt an opinion if the agent has no opinion at all, while the second threshold states the willingness to acquire a second opinion if the agent already has one. The goal is to add at most budget-many agents to the initial seed-sets such that the process started with these extended seed-sets stabilizes within the given number of rounds, with each agent having either both opinions or none. That is, our goal is to ensure that the spread of opinions is balanced. We show that the problem is NP-hard, and thus we study the problem from the perspective of parameterized complexity. In particular, we show that the problem is FPT when parameterized by the number of rounds, the maximum threshold, and the treewidth combined. This algorithm also applies to the combined parameter, the treedepth and the maximum threshold. Finally, we show that the problem is FPT when parameterized by the vertex cover number, the $3$-path vertex cover number, or the vertex integrity of the input network alone. To complement our tractability results, we show that the problem is W[1]-hard with respect to a) the sizes of the initial seed-sets and the feedback-vertex set number combined, even if all thresholds are bounded by a constant, and b) the budget, the 4-path vertex cover number, and the feedback-vertex set number combined, even if every activation process stabilizes in at most 4 rounds.

翻译：受著名的目标集选择问题启发，我们提出了一种新的离散模型，用于在社交网络中同时传播两种观点，并对其复杂性进行了初步研究。在该模型中，给定一个社交网络、每个观点的初始种子集、每个主体的两个阈值、一个预算以及轮次数量。第一个阈值表示主体在尚未持有任何观点时采纳某观点的意愿，而第二个阈值表示主体在已持有一种观点的情况下获取第二种观点的意愿。目标是向初始种子集中添加至多预算数量的主体，使得由这些扩展种子集启动的传播过程在给定轮次内稳定，且每个主体要么同时拥有两种观点，要么均不持有。即，目标是确保观点传播达到平衡。我们证明该问题是NP难的，因此从参数化复杂性的角度对其进行研究。特别地，我们表明当以轮次数量、最大阈值和树宽为联合参数时，该问题是固定参数可解（FPT）的。该算法同样适用于联合参数——树深度与最大阈值。最后，我们证明当仅以输入网络的顶点覆盖数、3-路径顶点覆盖数或顶点完整性为参数时，该问题也是FPT的。作为可处理性结果的补充，我们证明该问题是W[1]-难的，对于以下参数组合：a) 初始种子集大小与反馈顶点集数量的联合参数（即使所有阈值均有常数上界），以及b) 预算、4-路径顶点覆盖数与反馈顶点集数量的联合参数（即使每个激活过程至多在4轮内稳定）。