On the Computation of Fully Proportional Representation (1402.0580v1)

Abstract: We investigate two systems of fully proportional representation suggested by Chamberlin Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.

• The paper analyzes the computational complexity of determining winners in Chamberlin-Courant and Monroe systems for Fully Proportional Representation.
• Using parameterized complexity, it shows tractability often depends on parameters like the number of candidates, while remaining hard based on the number of winners, though most single-peaked cases are polynomial.
• The findings offer insights for designing computationally efficient electoral systems and lay a foundation for future AI algorithm development in collective decision-making.

Analyzing the Computational Complexity of Fully Proportional Representation

The paper "On the Computation of Fully Proportional Representation" by Betzler, Slinko, and Uhlmann provides an in-depth examination of two systems of fully proportional representation (FPR): the Chamberlin-Courant (CC) and Monroe systems. These systems are designed to ensure that every vote is represented in a way that minimizes the "sum of misrepresentations." However, determining winners in these systems is NP-hard, motivating the authors to explore parameterized complexity and discover instances where the problems become tractable.

The researchers introduce a new variation to the original problem formulations, the "minimax" rule, which aims to minimize the maximum misrepresentation. Despite this innovation, the paper concludes that the minimax variations remain NP-hard in the general case, similar to their classical counterparts.

To navigate around the computational obstacles, the paper meticulously investigates the parameterized complexity of winner determination for both classical and minimax systems. The analysis embeds diverse parameters such as the number of candidates, voters, and winners, offering a nuanced understanding of when these problems transition from intractable to tractable.

Key findings include the fixed-parameter tractability (FPT) of the problem when parameterized by the number of candidates, while parameterization by the number of winners consistently resulted in W[2]-hardness. Notably, for single-peaked electorates, most of the representation problems, barring the Monroe system, are solvable in polynomial time.

The implications of these results are significant for both theoretical exploration and practical applications. The computational complexity insights delineate boundary conditions under which polynomial-time solutions are feasible, potentially guiding political scientists in designing electoral systems that balance representativeness and computational efficiency. Moreover, the paper's theoretical framework sets a foundation for future AI development, particularly in designing efficient algorithms for complex decision-making processes involving consensus or collective choice.

Looking forward, these findings can stimulate further research in several directions. One avenue could involve refining the single-peakedness model or considering nearly single-peaked preferences, which may be more reflective of real-world scenarios. Furthermore, examining other forms of electoral manipulation or extending this paper to multiset models, where candidates can represent more than one voter group, could enrich our understanding of these voting mechanisms.

In conclusion, the paper meticulously explores the boundaries of computational complexity for fully proportional representation, unearthing tractable scenarios and presenting robust mathematical frameworks for further inquiry, both in the realms of algorithmic development and the design of fair, efficient representative systems.