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

Abstract


 
 
 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 (1989b) 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 in this setting the egalitarian version of this rule admits a polynomial-time winner determination algorithm. For the utilitarian version, we prove that winner determination remains NP-hard for the Borda scoring function; indeed, this hardness results extends to a large family of scoring functions. 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 then explore the power and limitations of this framework: we develop polynomial-time algorithms to find trees with the smallest maximum degree, diameter, or pathwidth, but show that it is NP-hard to check whether a given profile is single-peaked on a tree that is isomorphic to a given tree, or on a regular tree.
 
 

Preference Restrictions in Computational Social Choice: A Survey

“…Preferences Single-Peaked on Trees For preferences single-peaked on trees, the egalitarian version of the Chamberlin-Courant rule remains polynomial-time computable. The argument is similar to the interval-stabbing argument presented above, except that we now need to stab subtrees rather than subintervals (Peters et al, 2022).…”

“…While it is not as fast as the linear-time algorithm for preferences single-peaked on a line, it is very intuitive and easy to describe. The exposition in this section follows Peters et al (2022).…”

“…Theorem 4.20 (Peters et al, 2022). Suppose a profile P is single-peaked on some tree, and let T (P ) be the set of all such trees Then we can find in polynomial time an element of T (P ) with (i) a minimum number of leaves, (ii) a minimum number of internal vertices, (iii) a minimum diameter, (iv) a minimum pathwidth, (v) a minimum max-degree.…”

“…Theorem 4.21 (Peters et al, 2022). Given a profile P over A and a tree T with |V (T )| = |A|, it is NP-complete to decide whether the vertices of T can be labeled with alternatives so that P becomes single-peaked on that (labeled) tree.…”

“…The winner determination problem associated with the Chamberlin-Courant rule is usually translated into a decision problem by considering it as an optimization problem, so that we ask: "given a profile P , a target committee size k, and integer B, does there exist a committee W whose Chamberlin-Courant score is at least B?". This problem is NP-complete when using Borda scores (Lu and Boutilier, 2011; see also Peters et al, 2022). Thus, we will analyze restrictions of this problem to structured preference profiles.…”