We study the computational complexity of committee selection problem for several approval-based voting rules in the presence of outliers. Our first result shows that outlier consideration makes committee selection problem intractable for approval, net approval, and minisum approval voting rules. We then study parameterized complexity of this problem with five natural parameters, namely the target score, the size of the committee (and its dual parameter, the number of candidates outside the committee), the number of outliers (and its dual parameter, the number of non-outliers). For net approval and minisum approval voting rules, we provide a dichotomous result, resolving the parameterized complexity of this problem for all subsets of five natural parameters considered (by showing either FPT or W[1]-hardness for all subsets of parameters). For the approval voting rule, we resolve the parameterized complexity of this problem for all subsets of parameters except one. We also study approximation algorithms for this problem. We show that there does not exist any alpha(.) factor approximation algorithm for approval and net approval voting rules, for any computable function alpha(.), unless P=NP. For the minisum voting rule, we provide a pseudopolynomial (1+eps) factor approximation algorithm.
The fact that the closest string problem has similarities to the setup of approval ballots has been explored in the past, see, for instance @cite_26 . More specifically, and again in the spirit of observing similarities with the closest string family of problems, the minimax approval rule has been studied from the perspective of outliers @cite_32 . To the best of our knowledge, our work is the first comprehensive study of the complexity behavior of approval ballots in the presence of outliers.
@cite_26 @cite_32
2270513673 2181533063 1859600792 132136377
We study multiple referenda and committee elections, when the ballot of each voter is simply a set of approved binary issues (or candidates). Two well-known rules under this model are the commonly used candidate-wise majority, also called the minisum rule, as well as the minimax rule. In the former, the elected committee consists of the candidates approved by a majority of voters, whereas the latter picks a committee minimizing the maximum Hamming distance to all ballots. As these rules are in some ways extreme points in the whole spectrum of solutions, we consider a general family of rules, using the Ordered Weighted Averaging (OWA) operators. Each rule is parameterized by a weight vector, showing the importance of the i-th highest Hamming distance of the outcome to the voters. The objective then is to minimize the weighted sum of the (ordered) distances. We study mostly computational, but also manipulability properties for this family. We first exhibit that for many rules, it is NP-hard to find a winning committee. We then proceed to identify cases where the problem is either efficiently solvable, or approximable with a small approximation factor. Finally, we investigate the issue of manipulating such rules and provide conditions that make this possible. In this work, we initiate a detailed study of the parameterized complexity of Minimax Approval Voting. We demonstrate that the problem is W[2]-hard when parameterized by the size of the committee to be chosen, but does admit a FPT algorithm when parameterized by the number of strings that is more efficient than the previous ILP-based approaches for the problem. We also consider several combinations of parameters and provide a detailed landscape of the parameterized and kernelization complexity of the problem. We also study the version of the problem where we permit outliers, that is, where the chosen committee is required to satisfy a large number of voters (instead of all of them). In this context, we strengthen an APX-hardness result in the literature, and also show a simple but strong W-hardness result. We consider Approval Voting systems where each voter decides on a subset of candidates he she approves. We focus on the optimization problem of finding the committee of fixed size k, minimizing the maximal Hamming distance from a vote. In this paper we give a PTAS for this problem and hence resolve the open question raised by [AAAI’10]. The result is obtained by adapting the techniques developed by [JACM’02] originally used for the less constrained Closest String problem. The technique relies on extracting information and structural properties of constant size subsets of votes. We consider approval voting elections in which each voter votes for a (possibly empty) set of candidates and the outcome consists of a set of k candidates for some parameter k, e.g., committee elections. We are interested in the min-imax approval voting rule in which the outcome represents a compromise among the voters, in the sense that the maximum distance between the preference of any voter and the outcome is as small as possible. This voting rule has two main drawbacks. First, computing an outcome that minimizes the maximum distance is computationally hard. Furthermore, any algorithm that always returns such an outcome provides incentives to voters to misreport their true preferences. In order to circumvent these drawbacks, we consider approximation algorithms, i.e., algorithms that produce an outcome that approximates the minimax distance for any given instance. Such algorithms can be considered as alternative voting rules. We present a polynomial-time 2-approximation algorithm that uses a natural linear programming relaxation for the underlying optimization problem and deterministically rounds the fractional solution in order to compute the outcome; this result improves upon the previously best known algorithm that has an approximation ratio of 3. We are furthermore interested in approximation algorithms that are resistant to manipulation by (coalitions of) voters, i.e., algorithms that do not motivate voters to misreport their true preferences in order to improve their distance from the outcome. We complement previous results in the literature with new upper and lower bounds on strategyproof and group-strategyproof algorithms.