Recognizing and Eliciting Weakly Single Crossing Profiles on Trees
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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 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 preferences to it. We call this domain the weakly single crossing domain on trees. We present 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 profiles on trees even when we do not know any tree with respect to which the closure of the input profile is single crossing and the preferences can be queried only sequentially; moreover, the sequential order 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 preferences is large compared to the number of candidates, even if the input profile is known to be single crossing with respect to some given tree and the preferences can be accessed randomly.
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