On the Complexity of Constructive Control under Nearly Single-Peaked Preferences
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We investigate the complexity of Constructive Control by Adding/Deleting Votes (CCAV/CCDV) for r-approval, Condorcet, Maximin and Copeland^α in k-axes and k-candidates partition single-peaked elections. In general, we prove that CCAV and CCDV for most of the voting correspondences mentioned above are NP-hard even when k is a very small constant. Exceptions are CCAV and CCDV for Condorcet and CCAV for r-approval in k-axes single-peaked elections, which we show to be fixed-parameter tractable with respect to k. In addition, we give a polynomial-time algorithm for recognizing 2-axes elections, resolving an open problem. Our work leads to a number of dichotomy results. To establish an NP-hardness result, we also study a property of 3-regular bipartite graphs which may be of independent interest. In particular, we prove that for every 3-regular bipartite graph, there are two linear orders of its vertices such that the two endpoints of every edge are consecutive in at least one of the two orders.
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