Kemeny Elections with Bounded Single-peaked or Single-crossing Width

This paper is devoted to complexity results re-garding specific measures of proximity to single-peakedness and single-crossingness, called “single-peaked width ” [Cornaz et al., 2012] and “single-crossing width”. Thanks to the use of the PQ-tree data structure [Booth and Lueker, 1976], we show that both problems are polynomial time solvable in the general case (while it was only known for single-peaked width and in the case of narcissistic preferences). Furthermore, we establish one of the first results (to our knowledge) concerning the ef-fect of nearly single-peaked electorates on the com-plexity of an NP-hard voting system, namely we show the fixed-parameter tractability of Kemeny elections with respect to the parameters “single-peaked width ” and “single-crossing width”.