We consider distributed elections, where there is a center and k sites. In such distributed elections, each voter has preferences over some set of candidates, and each voter is assigned to exactly one site such that each site is aware only of the voters assigned to it. The center is able to directly communicate with each of the sites. We are interested in designing communication-efficient protocols, allowing the center to maintain (i.e., declare) a candidate which, with arbitrary high probability, is guaranteed to be a winner, or at least close to being a winner. We consider various single-winner voting rules, such as variants of Approval voting and scoring rules, tournament-based voting rules, and several round-based voting rules. For these voting rules, we show that, using communication which is logarithmic in the number of voters, it is possible for the center to maintain such approximate winners. We complement our protocols with lower bounds. Our results have implications in various scenarios, such as aggregating customer preferences in on-line shopping websites or supermarket chains and collecting votes from different polling stations of political elections.
In a recent paper, @cite_19 study winner determination for several multiwinner voting rules aiming at proportional representation. Dey and Narahari @cite_21 study sampling algorithms for estimating the margin of victory. These works deal with centralized elections, while the current paper considers distributed elections. Another paper worth mentioning in this context is the paper of @cite_7 which argues for the importance of developing fast communication-efficient protocols for computing winner in (centralized) streams; they also provide a simple sampling-based algorithm for approximating Borda winners.
@cite_19 @cite_21 @cite_7
2606550175 2110060186 2124837549 2168205737
We study a generic framework that provides a unified view on two important classes of problems: (i) extensions of the k-median problem where clients are interested in having multiple facilities in their vicinity (e.g., due to the fact that, with some small probability, the closest facility might be malfunctioning and so might not be available for using), and (ii) finding winners according to some appealing multiwinner election rules, i.e., election system aimed for choosing representatives bodies, such as parliaments, based on preferences of a population of voters over individual candidates. Each problem in our framework is associated with a vector of weights: we show that the approximability of the problem depends on structural properties of these vectors. We specifically focus on the harmonic sequence of weights for which the objective function interpreted in a multiwinner election setup reflects to the well-known Proportional Approval Voting (PAV) rule. Our main result is that, due to the specific (harmonic) structure of weights, the problem allows constant factor approximation. This is surprising since the problem can be interpreted as a variant of the k-median problem where we do not assume that the connection costs satisfy the triangle inequality. The algorithm we propose is based on dependent rounding [Srinivasan, FOCS'01] applied to the solution of a natural LP-relaxation of the problem. The rounding process is well known to produce distributions over integral solutions satisfying Negative Correlation (NC), which is usually sufficient for the analysis of approximation guarantees offered by rounding procedures. In our analysis, however, we need to use the fact that the carefully implemented rounding process satisfies a stronger property, called Negative Association (NA), which allows us to apply standard concentration bounds for conditional random variables. 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 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. 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.