The Degree of (Extended) Justified Representation and Its Optimization

2024-12-27

AI Summary

This paper studies the quantitative optimization of Justified Representation (JR) and Extended Justified Representation (EJR) in multi-winner approval voting: for each ℓ-cohesive group, the (E)JR degree $c$ is defined as the minimum number of approved winners among at least $c$ members of the group, and the goal is to maximize $c$. We introduce the MDJR and MDEJR optimization models; prove their NP-hardness and inapproximability within a factor of $(k/n)^{1-varepsilon}$; design a tight $(k/n)$-approximation algorithm; establish W[2]-hardness with respect to $c_{max}$; and provide a fixed-parameter tractable (FPT) algorithm parameterized by $c_{max}$. Our work integrates cohesive-group modeling, combinatorial optimization, and parameterized algorithm design, establishing a new computational paradigm for quantifying fairness in proportional representation.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Preferences
• Game Theory and Economic Paradigms: Social Choice / Voting

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Justified Representation (JR)/Extended Justified Representation (EJR) is a desirable axiom in multiwinner approval voting. In contrast to (E)JR only requires at least emph{one} voter to be represented in every cohesive group, we study its optimization version that maximizes the emph{number} of represented voters in each group. Given an instance, we say a winning committee provides an (E)JR degree of $c$ if at least $c$ voters in each $ell$-cohesive group have approved $ell$ winning candidates. Hence, every (E)JR committee provides the (E)JR degree of at least $1$. Besides proposing this new property, we propose the optimization problem of finding a winning committee that achieves the maximum possible (E)JR degree, called MDJR and MDEJR, corresponding to JR and EJR respectively. We study the computational complexity and approximability of MDJR of MDEJR. An (E)JR committee, which can be found in polynomial time, straightforwardly gives a $(k/n)$-approximation. On the other hand, we show that it is NP-hard to approximate MDJR and MDEJR to within a factor of $left(k/n ight)^{1-epsilon}$, for any $epsilon>0$, which complements the approximation. Next, we study the fixed-parameter-tractability of this problem. We show that both problems are W[2]-hard if $k$, the size of the winning committee, is specified as the parameter. However, when $c_{ ext{max}}$, the maximum value of $c$ such that a committee that provides an (E)JR degree of $c$ exists, is additionally given as a parameter, we show that both MDJR and MDEJR are fixed-parameter-tractable.

Problem

• Multi-winner Voting
• Proportional Representation
• Computational Complexity

Innovation

• Proportional Representation
• Fixed-Parameter Tractability
• Approximation Strategies