Efficient Algorithms for Monroe and CC Rules in Multi-Winner Elections with (Nearly) Structured Preferences

TL;DR

This paper presents efficient algorithms for winner determination in Monroe and Chamberlin-Courant rules under structured preferences, improving computational tractability in multi-winner elections with nearly single-peaked or single-crossing preferences.

Contribution

It introduces polynomial and fixed-parameter tractable algorithms for winner determination under structured preferences, addressing open complexity questions in the field.

Findings

Winner determination for Monroe rule is polynomial under single-crossing approval preferences.

Winner determination for both rules is FPT with respect to voter deletions to achieve structured preferences.

The results resolve several open complexity questions from prior literature.

Abstract

We investigate winner determination for two popular proportional representation systems: the Monroe and Chamberlin-Courant (abbrv. CC) systems. Our study focuses on (nearly) single-peaked resp. single-crossing preferences. We show that for single-crossing approval preferences, winner determination of the Monroe rule is polynomial, and for both rules, winner determination mostly admits FPT algorithms with respect to the number of voters to delete to obtain single-peaked or single-crossing preferences. Our results answer some complexity questions from the literature [18, 28, 21].