We consider the following problem: There is a set of items (e.g., movies) and a group of agents (e.g., passengers on a plane); each agent has some intrinsic utility for each of the items. Our goal is to pick a set of @math items that maximize the total derived utility of all the agents (i.e., in our example we are to pick @math movies that we put on the plane's entertainment system). However, the actual utility that an agent derives from a given item is only a fraction of its intrinsic one, and this fraction depends on how the agent ranks the item among the chosen, available, ones. We provide a formal specification of the model and provide concrete examples and settings where it is applicable. We show that the problem is hard in general, but we show a number of tractability results for its natural special cases.
Let us now move on to other related works and other related streams of research. Several known settings are recovered as particular cases of our general model. In particular, this applies to the case of the Chamberlin--Courant proportional representation rule @cite_2 , to the case of Proportional Approval Voting , and to (variants of) the budgeted social choice model @cite_25 @cite_23 @cite_42 . Computational complexity of the Chamberlin--Courant rule was first studied by , its parameterized complexity was analyzed by Betzler et al. , and the complexity under restricted domains was studied by Betzler et al. , @cite_8 , @cite_45 , and @cite_28 . The first approximation algorithm was proposed by Lu and Boutilier . The results on approximability were then extended in several directions by Skowron et al. . Proportional Approval Voting was studied computationally and axiomatically by @cite_35 @cite_37 and by Elkind and Lackner @cite_3 .
@cite_35 @cite_37 @cite_8 @cite_28 @cite_42 @cite_3 @cite_45 @cite_23 @cite_2 @cite_25
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Given a set of voters V, a set of candidates C, and voters' preferences over the candidates, multiwinner voting rules output a fixed-size subset of candidates committee. Under the Chamberlin---Courant multiwinner voting rule, one fixes a scoring vector of length |C|, and each voter's 'utility' for a given committee is defined to be the score that she assigns to her most preferred candidate in that committee; the goal is then to find a committee that maximizes the joint utility of all voters. The joint utility is typically identified either with the sum of all voters' utilities or with the utility of the least satisfied voter, resulting in, respectively, the utilitarian and the egalitarian variant of the Chamberlin---Courant's rule. For both of these cases, the problem of computing an optimal committee is NP-hard for general preferences, but becomes polynomial-time solvable if voters' preferences are single-peaked or single-crossing. In this paper, we propose a family of multiwinner voting rules that are based on the concept of ordered weighted average OWA and smoothly interpolate between the egalitarian and the utilitarian variants of the Chamberlin---Courant rule. We show that under moderate constraints on the weight vector we can recover many of the algorithmic results known for the egalitarian and the utilitarian version of Chamberlin---Courant's rule in this more general setting. 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 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 satisfaction or dissatisfaction of the voters with their representatives. The total (dis)satisfaction is calculated either as the sum of individual (dis)satisfactions in the utilitarian case or as the (dis)satisfaction of the worst off voter in 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 these rules 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 preference data. These experiments show that our simple and fast algorithms can, in many cases, find near-perfect solutions. We investigate two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.