# Proportional Representation under Single-Crossing Preferences Revisited

Andrei Constantinescu
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We have significantly improved the state of the art concerning the algorithmic complexity of the Chamberlin–Courant rule, both for preferences single-crossing on a line and for preferences single-crossing on a tree

Abstract:

We study the complexity of determining a winning committee under the Chamberlin--Courant voting rule when voters' preferences are single-crossing on a line, or, more generally, on a median graph (this class of graphs includes, e.g., trees and grids). For the line, Skowron et al. (2015) describe an $O(n^2mk)$ algorithm (where $n$, $m$, \$...More

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Introduction
Highlights
• The problem of computing election winners under various voting rules is perhaps the most fundamental research challenge in computational social choice (Brandt et al 2016)
• We reduce the Chamberlin–Courant winner determination problem to the well-studied directed acyclic graph (DAG) k-LINK PATH problem, and show that the instances of the latter problem that are produced by our reduction have the concave Monge property
• We focus on a family of multiwinner voting rules known as Chamberlin–Courant rules (Chamberlin and Courant 1983; Faliszewski et al 2017)
• We have significantly improved the state of the art concerning the algorithmic complexity of the Chamberlin–Courant rule, both for preferences single-crossing on a line and for preferences single-crossing on a tree
• We focused on the utilitarian version of the Chamberlin–Courant rule, where the goal is to minimize the sum of voters’ dissatisfactions; both of our O(nmk) algorithms can be modified to compute winners under the egalitarian version of this rule, where the goal is to minimize the dissatisfaction of the most misrepresented voter, by replacing ‘+’ with max in the respective dynamic programs
• This is no longer the case for our reduction to the k-LINK PATH problem (k-LPP) problem; by using binary search to reduce the egalitarian problem to the utilitarian problem, we can find solutions for the former in time O(nm log(n) log(nm)) using this approach
Results
• The authors have significantly improved the state of the art concerning the algorithmic complexity of the Chamberlin–Courant rule, both for preferences single-crossing on a line and for preferences single-crossing on a tree.
Conclusion
• Conclusions and Future Work

The authors have significantly improved the state of the art concerning the algorithmic complexity of the Chamberlin–Courant rule, both for preferences single-crossing on a line and for preferences single-crossing on a tree.
• The authors focused on the utilitarian version of the Chamberlin–Courant rule, where the goal is to minimize the sum of voters’ dissatisfactions; both of the O(nmk) algorithms can be modified to compute winners under the egalitarian version of this rule, where the goal is to minimize the dissatisfaction of the most misrepresented voter, by replacing ‘+’ with max in the respective dynamic programs
• This is no longer the case for the reduction to the k-LPP problem; by using binary search to reduce the egalitarian problem to the utilitarian problem, the authors can find solutions for the former in time O(nm log(n) log(nm)) using this approach
Summary
• ## Introduction:

The problem of computing election winners under various voting rules is perhaps the most fundamental research challenge in computational social choice (Brandt et al 2016)
• While this problem is typically easy for single-winner voting rules (with a few notable exceptions (Hemaspaandra, Hemaspaandra, and Rothe 1997; Rothe, Spakowski, and Vogel 2003)), for many rules that are supposed to return a set of winners, the winner determination problem is computationally demanding.
• These results extend to preferences that are nearly single-peaked or nearly single-crossing, for a suitable choice of distance measure (Cornaz, Galand, and Spanjaard 2012; Skowron et al 2015; Misra, Sonar, and Vaidyanathan 2017); see the survey by Elkind, Lackner, and Peters (2017) for a summary of results for restricted domains and the survey by Faliszewski et al (2017) for a discussion of other approaches to circumventing hardness results for the Chamberlin–Courant rule
• ## Results:

The authors have significantly improved the state of the art concerning the algorithmic complexity of the Chamberlin–Courant rule, both for preferences single-crossing on a line and for preferences single-crossing on a tree.
• ## Conclusion:

Conclusions and Future Work

The authors have significantly improved the state of the art concerning the algorithmic complexity of the Chamberlin–Courant rule, both for preferences single-crossing on a line and for preferences single-crossing on a tree.
• The authors focused on the utilitarian version of the Chamberlin–Courant rule, where the goal is to minimize the sum of voters’ dissatisfactions; both of the O(nmk) algorithms can be modified to compute winners under the egalitarian version of this rule, where the goal is to minimize the dissatisfaction of the most misrepresented voter, by replacing ‘+’ with max in the respective dynamic programs
• This is no longer the case for the reduction to the k-LPP problem; by using binary search to reduce the egalitarian problem to the utilitarian problem, the authors can find solutions for the former in time O(nm log(n) log(nm)) using this approach
Funding
• We have significantly improved the state of the art concerning the algorithmic complexity of the Chamberlin–Courant rule, both for preferences single-crossing on a line and for preferences single-crossing on a tree
Study subjects and analysis
voters: 3
Thus, we need to prove that each instance of CC-WINNER-SC is concave Monge. To this end, we will first argue that if there is an instance of CC-WINNER-SC that is not concave Monge, then there exists such an instance with three voters. Then we prove that every three-voter instance is concave Monge

voters: 3
Lemma 4. If there exists a non-concave Monge instance of CC-WINNER-SC, then there exists a non-concave Monge instance of CC-WINNER-SC with three voters. Proof

voters: 3
Proof. Consider a non-concave Monge instance of CCWINNER-SC with n = 3 voters. Note that n ≥ 4: indeed, for n < 3 there is no pair of vertices i, j that satisfies 0 < i + 1 < j < n

voters: 3
Proof. By Lemma 4, it suffices to consider instances with three voters. Thus, consider a three-voter profile that is single-crossing with respect to the voter order v1 v2 v3

Reference
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• We start by presenting the algorithm of Clearwater, Puppe, and Slinko (2015) (with a few typos corrected, and using our notation and terminology), and then show that its runtime can be exponential in the worst case, by exhibiting an explicit instance for which this occurs.
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