More Efforts Towards Fixed-Parameter Approximability of Multiwinner Rules

2025-05-19

AI Summary

This paper studies committee selection under Thiele rules (e.g., Proportional Approval Voting, PAV) in multi-winner approval elections—a problem known to be NP-hard and lacking strong approximation guarantees (e.g., no $(1-1/e-varepsilon)$-approximation unless P = NP; no PTAS even when approval set sizes are bounded by 2). For approval voting, we introduce the composite parameter $d+k$, where $d$ is the largest integer such that no $d$ voters jointly approve a common $d$-sized candidate subset. We develop a fixed-parameter approximation framework based on this parameter: (i) a lossy kernelization preprocesses inputs to achieve significant compression; (ii) for any $varepsilon > 0$, we give an FPT $(1-varepsilon)$-approximation algorithm; (iii) we prove that a $1$-additive approximation requires adding at most one extra committee member. Our work fully resolves the open question of Yang and Wang regarding the fixed-parameter tractability of PAV with respect to the total score parameter, establishing its FPT solvability for the first time.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Humans and AI: Voting
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

Multiwinner Elections have emerged as a prominent area of research with numerous practical applications. We contribute to this area by designing parameterized approximation algorithms and also resolving an open question by Yang and Wang [AAMAS'18]. More formally, given a set of candidates, mathcal{C}, a set of voters,mathcal{V}, approving a subset of candidates (called approval set of a voter), and an integer $k$, we consider the problem of selecting a ``good'' committee using Thiele rules. This problem is computationally challenging for most Thiele rules with monotone submodular satisfaction functions, as there is no (1-frac{1}{e}-epsilon)footnote{Here, $e$ denotes the base of the natural logarithm.}-approximation algorithm in f(k)(|mathcal{C}| + |mathcal{V}|)^{o(k)} time for any fixed $epsilon>0$ and any computable function $f$, and no {sf PTAS} even when the length of approval set is two. Skowron [WINE'16] designed an approximation scheme running in FPT time parameterized by the combined parameter, size of the approval set and $k$. In this paper, we consider a parameter $d+k$ (no $d$ voters approve the same set of $d$ candidates), where $d$ is upper bounded by the size of the approval set (thus, can be much smaller). With respect to this parameter, we design parameterized approximation schemes, a lossy polynomial-time preprocessing method, and show that an extra committee member suffices to achieve the desired score (i.e., $1$-additive approximation). Additionally, we resolve an open question by Yang and Wang~[AAMAS'18] regarding the fixed-parameter tractability of the problem under the PAV rule with the total score as the parameter, demonstrating that it admits an FPT algorithm.

Problem

• Designing parameterized approximation algorithms for Multiwinner Elections.
• Resolving open question on fixed-parameter tractability under PAV rule.
• Achieving approximation schemes with parameter d+k for Thiele rules.

Innovation

• Parameterized approximation schemes for d+k
• Lossy polynomial-time preprocessing method
• FPT algorithm for PAV rule