Gerrymandering: A Briber's Perspective

Abstract: We initiate the study of bribery problem in the context of gerrymandering and reverse gerrymandering. In our most general problem, the input is a set of voters having votes over a set of alternatives, a graph on the voters, a partition of voters into connected districts, cost of every voter for changing her district, a budget for the briber, and a favorite alternative of the briber. The briber needs to compute if the given partition can be modified so that (i) the favorite alternative of the briber wins the resulting election, (ii) the modification is budget feasible, and (iii) every new district is connected. We study four natural variants of the above problem -- the graph on voter being arbitrary vs complete graph (corresponds to removing connectedness requirement for districts) and the cost of bribing every voter being uniform vs non-uniform. We show that all the four problems are NP-complete even under quite restrictive scenarios. Hence our results show that district based elections are quite resistant under this new kind of electoral attack. We complement our hardness results with polynomial time algorithms in some other cases.