Definition: In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
Input: The domain of single crossing preference profiles is a widely studied domain in social choice theory. It has been generalized to the domain of single crossing preference profiles with respect to trees which inherits many desirable properties from the single crossing domain, for example, transitivity of majority relation, existence of polynomial time algorithms for finding winners of Kemeny voting rule, etc. In this paper, we consider a further generalization of the domain of single crossing preference profiles on trees to the domain consisting of all preference profiles which can be extended to single crossing preference profiles with respect to some tree by adding more preference orderings to it. We call this domain the weakly single crossing domain on trees. It is known that most of the desirable properties of the single crossing domain on trees continue to hold for the weakly single crossing domain on trees. In this paper, we prove several interesting structural properties of the weakly single crossing domain on trees and exploit them to design efficient algorithms for recognizing and eliciting weakly single crossing preference profiles on trees. In particular, we begin with showing that, for every weakly single crossing preference profile P on trees, there exists a unique inclusion-minimal set of preference orderings Q whose inclusion in P makes the resulting profile P = P ⊎Q single crossing with respect to some tree. We call the profile P the closure of P. Using this uniqueness of closure property of weakly single crossing preference profiles on trees and the known recognition algorithm of single crossing preference profiles on trees due to Clearwater et al. [CPS15], we design a polynomial time algorithm for recognizing weakly single crossing profiles on trees. We then move on to develop a polynomial time algorithm with low query complexity for eliciting weakly single crossing preference profiles on trees even when we do not know any tree with respect to which the closure of the input preference profile is single crossing and the preferences can be queried only sequentially; moreover, the sequential order of the input preference profile is also unknown. We complement the performance of our preference elicitation algorithm by proving that our algorithm makes an optimal number of queries up to constant factors when the number of preference orderings in the input preference profile is large compared to the number of candidates, even if the

Recognizing and Eliciting Weakly Single Crossing Profiles on Trees