In shift bribery, a briber seeks to promote his preferred candidate by paying voters to raise their ranking. Classical models of shift bribery assume voters act independently, overlooking the role of social influence. However, in reality, individuals are social beings and are often represented as part of a social network, where bribed voters may influence their neighbors, thereby amplifying the effect of persuasion. We study Shift bribery over Networks, where voters are modeled as nodes in a directed weighted graph, and arcs represent social influence between them. In this setting, bribery is not confined to directly targeted voters its effects can propagate through the network, influencing neighbors and amplifying persuasion. Given a budget and individual cost functions for shifting each voter's preference toward a designated candidate, the goal is to determine whether a shift strategy exists within budget that ensures the preferred candidate wins after both direct and network-propagated influence takes effect. We show that the problem is NP-Complete even with two candidates and unit costs, and W[2]-hard when parameterized by budget or maximum degree. On the positive side, we design polynomial-time algorithms for complete graphs under plurality and majority rules and path graphs for uniform edge weights, linear-time algorithms for transitive tournaments for two candidates, linear cost functions and uniform arc weights, and pseudo-polynomial algorithms for cluster graphs. We further prove the existence of fixed-parameter tractable algorithms with treewidth as parameter for two candidates, linear cost functions and uniform arc weights and pseudo-FPT with cluster vertex deletion number for two candidates and uniform arc weights. Together, these results give a detailed complexity landscape for shift bribery in social networks.

翻译：在顺位贿赂中，行贿者试图通过支付报酬让选民提升其偏好候选人的排名来促成该候选人当选。经典的顺位贿赂模型假设选民行为相互独立，忽略了社会影响的作用。然而现实中，个体是社会性存在，常被表示为社交网络的一部分，其中被贿赂的选民可能影响其邻居，从而放大说服效果。我们研究网络上的顺位贿赂问题，将选民建模为有向加权图中的节点，弧表示其间的社会影响。在此设定下，贿赂不仅限于直接目标选民，其效果可通过网络传播，影响邻居并放大说服效应。给定预算及针对每位选民向指定候选人调整偏好的个体成本函数，目标是确定是否存在一种在预算内的顺位调整策略，使得在直接及网络传播的影响生效后，偏好候选人能够获胜。我们证明即使只有两名候选人和单位成本，该问题也是NP完全的；当以预算或最大度作为参数时，该问题是W[2]-难的。在积极方面，我们针对完全图在多数决和过半数规则下设计了多项式时间算法，针对均匀边权的路径图设计了多项式时间算法，针对两名候选人、线性成本函数和均匀弧权的传递锦标赛设计了线性时间算法，并为团图设计了伪多项式算法。我们进一步证明对于两名候选人、线性成本函数和均匀弧权的情形，存在以树宽为参数的固定参数可解算法；对于两名候选人和均匀弧权的情形，存在以团顶点删除数为参数的伪固定参数可解算法。这些结果共同描绘了社交网络中顺位贿赂问题的完整计算复杂性图景。