# Proportional Representation under Single-Crossing Preferences Revisited

Weibo:

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

Code:

Data:

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

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

- Aggarwal, A.; Schieber, B.; and Tokuyama, T. 1994. Finding a minimum-weight k-link path in graphs with the concave Monge property and applications. Discrete and Computational Geometry 12: 263–280.
- Bein, W.; Larmore, L.; and Park, J. 199The d-edge shortest-path problem for a Monge graph. UNT Digital Library.
- Betzler, N.; Slinko, A.; and Uhlmann, J. 2011. On the Computation of Fully Proportional Representation. Journal of Artificial Intelligence Research 47.
- Brandt, F.; Conitzer, V.; Endriss, U.; Lang, J.; and Procaccia, A. D., eds. 2016. Handbook of Computational Social Choice. Cambridge University Press.
- Chamberlin, J.; and Courant, P. 1983. Representative Deliberations and Representative Decisions: Proportional Representation and the Borda Rule. American Political Science Review 77(3): 718–733.
- Clearwater, A.; Puppe, C.; and Slinko, A. 2015. Generalizing the Single-Crossing Property on Lines and Trees to Intermediate Preferences on Median Graphs. In IJCAI’15, 32–38.
- Cornaz, D.; Galand, L.; and Spanjaard, O. 2012. Bounded single-peaked width and proportional representation. In ECAI’12, 270–275.
- Cygan, M. 2012. Barricades. In Diks, K.; Idziaszek, T.; Lacki, J.; and Radoszewski, J., eds., Looking for a Challenge?, 63–67.
- Doignon, J.-P.; and Falmagne, J.-C. 1994. A polynomial time algorithm for unidimensional unfolding representations. Journal of Algorithms 16(2): 218–233.
- Elkind, E.; Lackner, M.; and Peters, D. 2017. Structured Preferences. In Endriss, U., ed., Trends in Computational Social Choice, chapter 10, 187–207. AI Access.
- Faliszewski, P.; Skowron, P.; Slinko, A.; and Talmon, N. 2017. Multiwinner Voting: A New Challenge for Social Choice Theory. In Endriss, U., ed., Trends in Computational Social Choice, chapter 2, 27–47. AI Access.
- Hemaspaandra, E.; Hemaspaandra, L. A.; and Rothe, J. 1997. Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP. Journal of the ACM 44(6): 806–825.
- Kung, F.-C. 2015. Sorting out single-crossing preferences on networks. Social Choice and Welfare 44(3): 663–672.
- Lu, T.; and Boutilier, C. 2011. Budgeted Social Choice: From Consensus to Personalized Decision Making. In Proceedings of IJCAI’11, 280–286.
- Misra, N.; Sonar, C.; and Vaidyanathan, P. 2017. On the complexity of Chamberlin-Courant on almost structured profiles. In ADT’17, 124–138. Springer.
- Peters, D.; and Elkind, E. 20Preferences Single-Peaked on Nice Trees. In AAAI’16, 594–600.
- Peters, D.; Yu, L.; Chan, H.; and Elkind, E. 2020. Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results. CoRR abs/2007.06549.
- Procaccia, A.; Rosenschein, J.; and Zohar, A. 2008. On the complexity of achieving proportional representation. Social Choice and Welfare 30: 353–362.
- Puppe, C.; and Slinko, A. 20Condorcet domains, median graphs and the single-crossing property. Economic Theory 67(1): 285–318.
- Rothe, J.; Spakowski, H.; and Vogel, J. 2003. Exact complexity of the winner problem for Young elections. Theory of Computing Systems 36(4): 375–386.
- Schieber, B. 1995. Computing a minimum-weight k-link path in graphs with the concave Monge property. J. Algorithms 29: 204–222.
- Skowron, P.; Yu, L.; Faliszewski, P.; and Elkind, E. 2015. The complexity of fully proportional representation for single-crossing electorates. Theoretical Computer Science 569: 43–57.
- Yu, L.; Chan, H.; and Elkind, E. 2013. Multiwinner Elections Under Preferences That Are Single-Peaked on a Tree. In IJCAI’13, 425–431.
- 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.

Full Text

Tags

Comments