On Recognising Nearly Single-Crossing Preferences

Abstract

If voters' preferences are one-dimensional, many hard problems in computational social choice become tractable. A preference profile can be classified as one-dimensional if it has the single-crossing property, which requires that the voters can be ordered from left to right so that their preferences are consistent with this order. In practice, preferences may exhibit some one-dimensional structure, despite not being single-crossing in the formal sense. Hence, we ask whether one can identify preference profiles that are close to being single-crossing. We consider three distance measures, which are based on partitioning voters or candidates or performing a small number of swaps in each vote. We prove that it can be efficiently decided if voters can be split into two single-crossing groups. Also, for every fixed k >= 1 we can decide in polynomial time if a profile can be made single-crossing by performing at most k candidate swaps per vote. In contrast, for each k >= 3 it is NP-complete to decide whether candidates can be partitioned into k sets so that the restriction of the input profile to each set is single-crossing.

Measuring Nearly Single-Peakedness of an Electorate: Some New Insights

“…Besides, a local version of the new measure (where, instead of the sum, we minimize the maximum over the voters of the number of forbidden triples), together with a comparison to local versions of V D and GS, might be investigated. Also, it would be interesting to undertake the same type of approach, based on the very definition of a restricted domain, for defining distance measures to other domains, such as the single-crossing domain [16], or single-peaked preferences on a graph [18].…”

Restricted Domains of Dichotomous Preferences with Possibly Incomplete Information

“…Lastly, a different-yet intuitively related-task on domain restrictions, pioneered by Faliszewski, Hemaspaandra, and Hemaspaandra (2014) and so far only explored on domains of total orders, is about recognising profiles that nearly enjoy a given structure (Elkind and Lackner 2014;Erdélyi, Lackner, and Pfandler 2017;Jaeckle, Peters, and Elkind 2018), according to some distance metric. This is very natural area to investigate for dichotomous preferences as well, but this is a topic for another paper.…”