The control complexity of r-approval: from the single-peaked case to the general case

Abstract

A natural generalization of the single-peaked elections is the k-peaked elections, where at most k peaks are allowed in each vote. Motivated by NP-hardness in general and polynomial-time solvability in single-peaked elections, we aim at establishing a complexity dichotomy of several control problems for r-Approval voting in k-peaked elections with respect to k. It turns out that most NP-completeness results in general also hold in k-peaked elections, even for k=2,3. On the other hand, we derive polynomial-time algorithms for certain control problems for k=2. In addition, we also study the problems from the viewpoint of parameterized complexity and achieve both FPT and W-hardness results. Several of our results apply to approval voting and sincere-strategy preference-based approval voting as well.