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

@article{Peters2022PreferencesSO, title={Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results}, author={Dominik Peters and Lan Yu and Hau Chan and Edith Elkind}, journal={J. Artif. Intell. Res.}, year={2022}, volume={73}, pages={231-276} }

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…

## 2 Citations

### Proportional Representation under Single-Crossing Preferences Revisited

- Computer ScienceAAAI
- 2021

It is shown that a simple tweak improves the time complexity to O(nmk) and is improved this bound for k=Ω(log n) by reducing the problem to the k-link path problem for DAGs with concave Monge weights, obtaining a nm2^O(√(log k log log n)) algorithm for the general case.

### Choosing Fair Committees and Budgets: Proportionality in Multi-Winner Elections and Participatory Budgeting

- Economics
- 2021

Participatory budgeting is a way to allow citizens to have a say in deciding how to spend public funds. It is a generalisation of multi-winner voting, where a committee is to be elected based on the…

## References

SHOWING 1-10 OF 79 REFERENCES

### Multiwinner Elections Under Preferences That Are Single-Peaked on a Tree

- Computer ScienceIJCAI
- 2013

It is proved that the complexity of electing a committee under several variants of the Chamberlin-Courant rule when the voters' preferences are single-peaked on a tree remains computationally hard on trees that have bounded degree, diameter, or pathwidth.

### Preferences Single-Peaked on Nice Trees

- Computer ScienceAAAI
- 2016

It is NP-hard to decide whether an input profile is single-peaked with respect to a given tree, and this framework is used to identify a new class of profiles that admit an efficient algorithm for a popular variant of the Chamberlin-Courant rule.

### Recognizing Single-Peaked Preferences on an Arbitrary Graph: Complexity and Algorithms

- Computer Science, MathematicsSAGT
- 2020

If the optimal number of edges is m-1 (where m is the number of candidates) then any optimal solution of the ILP is integer and thus the integrality constraints can be relaxed, providing an alternative proof of the polynomial-time complexity of recognizing single-peaked preferences on a tree.

### Preferences Single-Peaked on a Circle

- Computer ScienceAAAI
- 2017

This work proves that Proportional Approval Voting can be computed in polynomial time for profiles that are single-peaked on a circle, and gives a fast recognition algorithm of this domain, provides a characterisation by finitely many forbidden subprofiles, and shows that many popular single- and multi-winner voting rules are polynometric-time computable on this domain.

### Preferences Single-Peaked on a Tree: Sampling and Tree Recognition

- EconomicsIJCAI
- 2019

This work develops a procedure to uniformly sample preferences that are singlepeaked on a given tree, and develops a solution to the problem of reliably identifying the tree that generated the observed votes.

### Bounded Single-Peaked Width and Proportional Representation

- EconomicsECAI
- 2012

This paper establishes the polynomiality of determining the single-peaked width of a preference profile (minimum width for a partition of candidates into clusters compatible with clustered single- peakedness), and shows that the proportional representation (PR) problem becomesPolynomial when the size of the largest cluster of candidates (width) is bounded.

### On the Computation of Fully Proportional Representation

- Computer Science, MathematicsJ. Artif. Intell. Res.
- 2013

This work investigates two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe and investigates the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters.

### Eliciting single-peaked preferences using comparison queries

- Economics, Computer ScienceAAMAS '07
- 2007

It is shown that single-peaked preferences can be elicited using only a linear number of comparison queries, if either the order with respect to which preferences are single- peaked is known, or at least one other agent's complete preferences are known.