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.
In this paper we give a PTAS for the Minimax Approval Voting problem. Our work is based on the PTAS for Closest String @cite_1 , which is a similar problem to @math but there we do not have the restriction on the number of 1's in the result. Technically, our contribution is the method of handling the number of 1's in the output. We also believe that our presentation is somewhat more intuitive.
@cite_1
1859600792 2181533063 132136377 2963006872
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. 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 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. In the Closest String problem one is given a family S of equal-length strings over some fixed alphabet, and the task is to find a string y that minimizes the maximum Hamming distance between y and a string from S. While polynomial-time approximation schemes (PTASes) for this problem are known for a long time [; J. ACM'02], no efficient polynomial-time approximation scheme (EPTAS) has been proposed so far. In this paper, we prove that the existence of an EPTAS for Closest String is in fact unlikely, as it would imply that FPT=W[1], a highly unexpected collapse in the hierarchy of parameterized complexity classes. Our proof also shows that the existence of a PTAS for Closest String with running time f(eps) n^o(1 eps), for any computable function f, would contradict the Exponential Time Hypothesis.