On the computational complexity of variants of combinatorial voter control in elections

Voter control problems model situations in which an exter-nal agent tries to affect the result of an election by adding or deleting the fewest number of voters. The goal of the agent is to make a specific can-didate either win (constructive control) or lose (destructive control) the election. We study the constructive and destructive voter control prob-lems when adding and deleting voters have a combinatorial flavor: If we add (resp. delete) a voter v, we also add (resp. delete) a bundle κ(v) of voters that are associated with v. While the bundle κ(v) may have more than one voter, a voter may also be associated with more than one voter. We analyze the computational complexity of the four voter control problems for the Plurality rule. We obtain that, in general, making a candidate lose is computation-ally easier than making her win. In particular, if the bundling relation is symmetric (i.e. ∀w: w ∈ κ(v) ⇔ v ∈ κ(w)), and if each voter has at most two voters associated with him, then destructive control is polynomial-time solvable while the constructive variant remains NP-hard. Even if the bundles are disjoint (i.e. ∀w: w ∈ κ(v) ⇔ κ(v) = κ(w)), the con-structive problem variants remain intractable. Finally, the minimization variant of constructive control by adding voters does not admit an effi-cient approximation algorithm, unless P = NP.