Shift Bribery over Social Networks

TL;DR

This work studies Shift Bribery over Social Networks, blending bribery and influence diffusion by modeling voters as nodes in a directed weighted graph with shift-based cost functions. It maps the problem's computational complexity across graph classes (general, complete, cluster, path, bipartite) and under different rules (majority, plurality), revealing NP-completeness and W[2]-hardness in general, alongside polynomial-time and fixed-parameter tractable results for structured graphs and parameters such as treewidth and cluster deletion number. A key methodological contribution is a dynamic programming framework over tree decompositions that yields FPT results for two candidates with linear costs and unit weights, plus pseudo-polynomial and kernel-like approaches for other settings. The findings delineate a detailed complexity landscape, advancing understanding of when network-aware bribery can be solved efficiently and guiding practical algorithm design for networked elections. The results have implications for designing robust election mechanisms and for developing targeted strategies that account for social diffusion in campaign planning.

Abstract

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.