We study the complexity of (approximate) winner determination under the Monroe and Chamberlin--Courant multiwinner voting rules, which determine the set of representatives by optimizing the total (dis)satisfaction of the voters with their representatives. The total (dis)satisfaction is calculated either as the sum of individual (dis)satisfactions (the utilitarian case) or as the (dis)satisfaction of the worst off voter (the egalitarian case). We provide good approximation algorithms for the satisfaction-based utilitarian versions of the Monroe and Chamberlin--Courant rules, and inapproximability results for the dissatisfaction-based utilitarian versions of them and also for all egalitarian cases. Our algorithms are applicable and particularly appealing when voters submit truncated ballots. We provide experimental evaluation of the algorithms both on real-life preference-aggregation data and on synthetic data. These experiments show that our simple and fast algorithms can in many cases find near-perfect solutions.
The work of @cite_4 and of @cite_17 on approximate winner determination for Dodgson's rule is particularly interesting from our perspective. In the former, the authors advocate treating approximation algorithms for Dodgson's rule as voting rules in their own right and design them to have desirable properties. In the latter, the authors show that a well-established voting rule (so-called Maximin rule) is a reasonable (though not optimal) approximation of Dodgson's rule. This perspective is important for anyone interested in using approximation algorithms for winner determination in elections (as might be the case for our algorithms for the Monroe and Chamberlin--Courant rules).
@cite_4 @cite_17
2024298276 2110060186 1516798378 2017658482
The voting rules proposed by Dodgson and Young are both designed to find an alternative closest to being a Condorcet winner, according to two different notions of proximity; the score of a given alternative is known to be hard to compute under either rule. In this paper, we put forward two algorithms for approximating the Dodgson score: a combinatorial, greedy algorithm and an LP-based algorithm, both of which yield an approximation ratio of H"m"-"1, where m is the number of alternatives and H"m"-"1 is the (m-1)st harmonic number. We also prove that our algorithms are optimal within a factor of 2, unless problems in NP have quasi-polynomial-time algorithms. Despite the intuitive appeal of the greedy algorithm, we argue that the LP-based algorithm has an advantage from a social choice point of view. Further, we demonstrate that computing any reasonable approximation of the ranking produced by Dodgson@?s rule is NP-hard. This result provides a complexity-theoretic explanation of sharp discrepancies that have been observed in the social choice theory literature when comparing Dodgson elections with simpler voting rules. Finally, we show that the problem of calculating the Young score is NP-hard to approximate by any factor. This leads to an inapproximability result for the Young ranking. 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. We then consider elections that are both single-peaked and single-crossing, and develop an efficient algorithm for the egalitarian variant of Monroe's rule for such elections. While [3] have recently presented a polynomial-time algorithm for this rule under single-peaked preferences, our algorithm has considerably better worst-case running time than that of In the year 1876 the mathematician Charles Dodgson, who wrote fiction under the now more famous name of Lewis Carroll, devised a beautiful voting system that has long fascinated political scientists. However, determining the winner of a Dodgson election is known to be complete for the Θ2 p level of the polynomial hierarchy. This implies that unless P=NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates—although the number of voters may still be polynomial in the number of candidates—a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it “knows” that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner. In 1876 Charles Lutwidge Dodgson suggested the intriguing voting rule that today bears his name. Although Dodgson's rule is one of the most well-studied voting rules, it suffers from serious deficiencies, both from the computational point of view - it is NP-hard even to approximate the Dodgson score within sublogarithmic factors - and from the social choice point of view - it fails basic social choice desiderata such as monotonicity and homogeneity. In a previous paper [, SODA 2009] we have asked whether there are approximation algorithms for Dodgson's rule that are monotonic or homogeneous. In this paper we give definitive answers to these questions. We design a monotonic exponential-time algorithm that yields a 2-approximation to the Dodgson score, while matching this result with a tight lower bound. We also present a monotonic polynomial-time O(log m)-approximation algorithm (where m is the number of alternatives); this result is tight as well due to a complexity-theoretic lower bound. Furthermore, we show that a slight variation of a known voting rule yields a monotonic, homogeneous, polynomial-time O(m log m)-approximation algorithm, and establish that it is impossible to achieve a better approximation ratio even if one just asks for homogeneity. We complete the picture by studying several additional social choice properties; for these properties, we prove that algorithms with an approximation ratio that depends only on m do not exist.