Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results

by   Dominik Peters, et al.

A preference profile is single-peaked 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 single-peaked on a tree. We study the complexity of multiwinner elections under several variants of the Chamberlin-Courant rule for preferences single-peaked on trees. We show that the egalitarian version of this problem admits a polynomial-time algorithm. For the utilitarian version, we prove that winner determination remains NP-hard, 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 single-peaked 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 single-peaked. 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 single-peaked. We also consider several other optimization criteria for trees: for some we obtain polynomial-time algorithms, while for others we show NP-hardness results.


page 1

page 2

page 3

page 4


Recognizing and Eliciting Weakly Single Crossing Profiles on Trees

The domain of single crossing preference profiles is a widely studied do...

Explaining Preferences by Multiple Patterns in Voters' Behavior

In some preference aggregation scenarios, voters' preferences are highly...

Polynomial tuning of multiparametric combinatorial samplers

Boltzmann samplers and the recursive method are prominent algorithmic fr...

New Algorithms for Unordered Tree Inclusion

The tree inclusion problem is, given two node-labeled trees P and T (the...

Incomplete Preferences in Single-Peaked Electorates

Incomplete preferences are likely to arise in real-world preference aggr...

Determining a Slater Winner is Complete for Parallel Access to NP

We consider the complexity of deciding the winner of an election under t...

The Smallest Hard Trees

We find an orientation of a tree with 20 vertices such that the correspo...