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We investigate the complexity of $r$-Approval control problems in $k$-peakedelections, where at most $k$ peaks are allowed in each vote with respect to anorder of the candidates. We show that most NP-hardness results in generalelections also hold in k-peaked elections even for $k=2,3$. On the other hand,we derive polynomial-time algorithms for some problems for $k=2$. All ourNP-hardness results apply to Approval and sincere-strategy preference-basedApproval as well. Our study leads to many dichotomy results for the problemsconsidered in this paper, with respect to the values of $k$ and $r$. Inaddition, we study $r$-Approval control problems from the viewpoint ofparameterized complexity and achieve both fixed-parameter tractability resultsand W-hardness results, with respect to the solution size. Along the wayexploring the complexity of control problems, we obtain two byproducts whichare of independent interest. First, we prove that every graph of maximum degree3 admits a specific 2-interval representation where every 2-intervalcorresponding to a vertex contains a trivial interval and, moreover,2-intervals may only intersect at the endpoints of the intervals. Second, wedevelop a fixed-parameter tractable algorithm for a generalized $r$-Set Packingproblem with respect to the solution size, where each element in the givenuniversal set is allowed to occur in more than one r-subset in the solution.

The Control Complexity of $r$-Approval: from the Single-Peaked Case to the General Case