Abstract
Justified representation (JR) is a standard notion of representation in multiwinner approval voting. Not only does a JR committee always exist, but previous work has also shown through experiments that the JR condition can typically be fulfilled by groups of fewer than k candidates, where k is the target size of the committee. In this paper, we study such groups—known as n/k-justifying groups—both theoretically and empirically. First, we show that under the impartial culture model, n/k-justifying groups of size less than k/2 are likely to exist, which implies that the number of JR committees is usually large. We then present efficient approximation algorithms that compute a small n/k-justifying group for any given instance, and a polynomial-time exact algorithm when the instance admits a tree representation. In addition, we demonstrate that small n/k-justifying groups can often be useful for obtaining a gender-balanced JR committee even though the problem is NP-hard.
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Notes
- 1.
In the full version of our paper [5], we extend this hardness to the case \(n/k > 1\).
- 2.
Bredereck et al. [3] studied maximizing objective functions of committees subject to gender balance and other diversity constraints, but did not consider JR.
- 3.
The condition that every candidate is approved by at least one voter is necessary. Indeed, if the last approval set in Example 1 is changed from \(\{c_k,c_{k+1},\dots ,c_m\}\) to \(\{c_k\}\), then there is only one JR committee: \(\{c_1,c_2,\dots ,c_k\}\).
- 4.
Recall that a running time is said to be quasi-polynomial if it is of the form \(\exp (\log ^{O(1)} I)\), where I denotes the input size (in our case, \(I = (nm)^{O(1)}\)).
- 5.
- 6.
In particular, we refer to the work of Elkind et al. [6] for motivation of the Euclidean models.
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Acknowledgments
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 101002854), from the Deutsche Forschungsgemeinschaft under grant BR 4744/2-1, from JST PRESTO under grant number JPMJPR20C1, from the Singapore Ministry of Education under grant number MOE-T2EP20221-0001, and from an NUS Start-up Grant. We would like to thank the anonymous SAGT reviewers for their comments.
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Elkind, E., Faliszewski, P., Igarashi, A., Manurangsi, P., Schmidt-Kraepelin, U., Suksompong, W. (2022). Justifying Groups in Multiwinner Approval Voting. In: Kanellopoulos, P., Kyropoulou, M., Voudouris, A. (eds) Algorithmic Game Theory. SAGT 2022. Lecture Notes in Computer Science, vol 13584. Springer, Cham. https://doi.org/10.1007/978-3-031-15714-1_27
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