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

Single-peaked elections have been attracting much attention recently. It turned out that manyNPhard voting problems become polynomial-time solvable when restricted to single-peaked elections. A natural generalization of the single-peaked elections is the k-peaked elections, where at most k peaks are allowed in each vote in the election. In this paper, we mainly aim at establishing a complexity dichotomy of controlling behaviors of a variant of the sincere-strategy preference-based approval voting in k-peaked elections for different values of k. It turns out that mostNP-hardness results in the general case also hold in k-peaked elections, even for k = 2, 3. On the other hand, we derive polynomial-time algorithms for certain sincere-strategy preference-based approval voting control problems for k = 2. In addition, we also study the sincere-strategy preference-based approval control problems from the viewpoint of parameterized complexity and prove some FPT and W-hardness results.