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arXiv · 1307.1252

The Complexity of Fully Proportional Representation for Single-Crossing Electorates

Piotr Skowron·Lan Yu·Piotr Faliszewski·Edith Elkind
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Abstract

We study the complexity of winner determination in single-crossing elections under two classic fully proportional representation rules---Chamberlin--Courant's rule and Monroe's rule. Winner determination for these rules is known to be NP-hard for unrestricted preferences. We show that for single-crossing preferences this problem admits a polynomial-time algorithm for Chamberlin--Courant's rule, but remains NP-hard for Monroe's rule. Our algorithm for Chamberlin--Courant's rule can be modified to work for elections with bounded single-crossing width. To circumvent the hardness result for Monroe's rule, we consider single-crossing elections that satisfy an additional constraint, namely, ones where each candidate is ranked first by at least one voter (such elections are called narcissistic). For single-crossing narcissistic elections, we provide an efficient algorithm for the egalitarian version of Monroe's rule.

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Piotr Skowron, Lan Yu, Piotr Faliszewski, Edith Elkind. 2013-07-04. The Complexity of Fully Proportional Representation for Single-Crossing Electorates. https://arxiv.org/abs/1307.1252

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