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Published in:
2013 | OriginalPaper | Chapter
The Complexity of Fully Proportional Representation for Single-Crossing Electorates
Authors : Piotr Skowron, Lan Yu, Piotr Faliszewski, Edith Elkind
Published in: Algorithmic Game Theory
Publisher: Springer Berlin Heidelberg
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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.