Priced Gerrymandering

We initiate the study of the gerrymandering problem when changing the district of a voter incurs a certain cost. In this 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, a cost of every voter for changing her district, a budget, and a target winner. We need to compute if the given partition can be modified so that (i) the target alternative 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 the voters being arbitrary vs complete graph (corresponds to removing the connectivity requirement for districts) and the cost of moving 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 intractability results by showing that two of our problems admit polynomial-time algorithms if the budget or the number of districts is a constant. We believe that our findings would help authorities to impose certain constraints, connectedness of districts for example, to make gerrymandering computationally harder.