Preferences SinglePeaked on a Tree: Multiwinner Elections and Structural Results
Abstract
A preference profile is singlepeaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989) described an efficient algorithm for deciding if a given profile is singlepeaked on a tree. We study the complexity of multiwinner elections under several variants of the ChamberlinCourant rule for preferences singlepeaked on trees. We show that the egalitarian version of this problem admits a polynomialtime algorithm. For the utilitarian version, we prove that winner determination remains NPhard, even for the Borda scoring function; however, a winning committee can be found in polynomial time if either the number of leaves or the number of internal vertices of the underlying tree is bounded by a constant. To benefit from these positive results, we need a procedure that can determine whether a given profile is singlepeaked on a tree that has additional desirable properties (such as, e.g., a small number of leaves). To address this challenge, we develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is singlepeaked. We show how to use this representation to efficiently find the best tree for a given profile for use with our winner determination algorithms: Given a profile, we can efficiently find a tree with the minimum number of leaves, or a tree with the minimum number of internal vertices among trees on which the profile is singlepeaked. We also consider several other optimization criteria for trees: for some we obtain polynomialtime algorithms, while for others we show NPhardness results.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.06549
 Bibcode:
 2020arXiv200706549P
 Keywords:

 Computer Science  Computer Science and Game Theory
 EPrint:
 44 pages, extends works published at AAAI 2016 and IJCAI 2013, published version