Recognizing and eliciting weakly single crossing profiles on trees

Single crossing profiles on trees are not downward closed — a sub-profile of a single crossing profile on trees is not necessarily a single crossing profile on trees. We define weakly single-crossing profiles on trees to be all single crossing profiles on trees and their sub-profiles thereby restoring downward closedness. We design a polynomial-time algorithm for recognizing these profiles. We then develop an efficient elicitation algorithm for this domain which works even if the preferences can be accessed only sequentially and the underlying single-crossing tree structure is not known beforehand. We complement our algorithmic results by proving a matching lower bound on the query complexity of our elicitation algorithm when the number of voters is large compared to the number of candidates. We also prove a lower bound of $\Omega (m^2\log n)$ on the number of queries that any algorithm needs to ask to elicit single-crossing profile when random queries are allowed. This resolves an open question in [18] and proves optimality of their preference elicitation algorithm when random queries are allowed.