Abstract
In a multiwinner election based on the Condorcet criterion, we are given a set of candidates, and a set of voters with strict preference rankings over the candidates. A committee is weakly Gehrlein stable (WGS) if each committee member is preferred to each non-member by at least half of the voters. Recently, Aziz et al. [IJCAI 2017] studied the computational complexity of finding a WGS committee of size k. They show that this problem is NP-hard in general and polynomial-time solvable when the number of voters is odd. In this article, we initiate a systematic study of the problem in the realm of parameterized complexity. We first show that the problem is W[1]-hard when parameterized by the size of the committee. To overcome this intractability result, we use a known reformulation of WGS as a problem on directed graphs and then use parameters that measure the “structure” of these directed graphs. We show that the problem is fixed parameter tractable and admits linear kernels with respect to these parameters; and also present an exact-exponential time algorithm with running in time \({\mathcal {O}}(1.2207^nn^{{\mathcal {O}}(1)})\), where n denotes the number of candidates.
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Notes
Throughout the paper, we considered strict preferences. However, all the algorithms can be used even in the case of weak preference order (voters do not have strict ranking over all the candidates). In the case of weak preference order, we say that a candidate c wins over candidate \(c'\), if more voters prefer c over \(c'\) in the pairwise election between c and \(c'\).
A strongly connected component is maximal by definition as it contains all those vertices that are connected be a directed path between every pair.
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A preliminary version of this article appeared in the proceedings of International Conference on Autonomous Agents and Multiagent Systems (AAMAS) 2019. This work was supported by SERB-NPDF fellowship (PDF/2016/003508) of DST, India; ISF (1176/18); European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant No. 819416), and Swarnajayanti Fellowship Grant DST/SJF/MSA- 01/2017-18.
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Gupta, S., Jain, P., Roy, S. et al. Gehrlein stability in committee selection: parameterized hardness and algorithms. Auton Agent Multi-Agent Syst 34, 27 (2020). https://doi.org/10.1007/s10458-020-09452-z
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DOI: https://doi.org/10.1007/s10458-020-09452-z