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Even More Effort Towards Improved Bounds and Fixed-Parameter Tractability for Multiwinner Rules

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Abstract

Multiwinner elections have proven to be a fruitful research topic with many real-world applications. We contribute to this line of research by improving the state of the art regarding the computational complexity of computing good committees. More formally, given a set of candidates \(\mathcal{C}\), a set of voters \(\mathcal{V}\)—each ranking the candidates according to their preferences, and an integer k; a multiwinner voting rule identifies a k-sized committee, based on these given voter preferences. In this paper we consider several utilitarian and egailitarian ordered weighted average scoring rules, which are an extensively-researched family of rules (and a subfamily of the family of committee scoring rules). First, we improve the result of Betzler et al. (JAIR 47:475–519, 2013), which gave a \({\mathcal {O}}(n^n)\) algorithm for computing winner under the Chamberlin–Courant rule, where n is the number of voters; to a running time of \({\mathcal {O}}(2^n)\), which is optimal. Furthermore, we study the parameterized complexity of the Pessimist voting rule and describe a few tractable and intractable cases. Apart from such utilitarian voting rules, we extend our study and consider egalitarian median and egalitarian mean (both committee scoring rules), showing some tractable and intractable results, based on nontrivial structural observations.

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Notes

  1. Indeed, there might be several co-winning committees; we ignore such issues of tie-breaking here as they only clutter the technical presentation.

  2. This can be achieved by adding some new elements in the universe and sets in the family appropriately.

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Authors and Affiliations

  1. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Sushmita Gupta & Saket Saurabh

  2. Indian Institute of Technology Jodhpur, Jodhpur, India

    Pallavi Jain

  3. University of Bergen, Bergen, Norway

    Saket Saurabh

  4. Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Nimrod Talmon

Authors
  1. Sushmita Gupta
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  2. Pallavi Jain
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  3. Saket Saurabh
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  4. Nimrod Talmon
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Correspondence to Pallavi Jain.

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A preliminary version of this paper appeared in the 30th International Joint Conference on Artificial Intelligence (IJCAI-21).

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Gupta, S., Jain, P., Saurabh, S. et al. Even More Effort Towards Improved Bounds and Fixed-Parameter Tractability for Multiwinner Rules. Algorithmica 85, 3717–3740 (2023). https://doi.org/10.1007/s00453-023-01155-7

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  • DOI: https://doi.org/10.1007/s00453-023-01155-7

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