Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results
Dominik Peters, Lan Yu, Hau Chan, Edith Elkind

TL;DR
This paper explores the computational complexity of multiwinner election rules under preferences that are single-peaked on a tree, providing algorithms and hardness results for various scenarios and structural properties.
Contribution
It introduces a structural approach to efficiently identify trees with specific properties for preferences single-peaked on a tree, and analyzes winner determination complexity under these conditions.
Findings
Egalitarian Chamberlin-Courant winner determination is polynomial-time solvable.
Utilitarian winner determination is NP-hard, but polynomial for bounded leaves or internal vertices.
Structural representation enables efficient optimization of tree properties for preferences.
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 (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…
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