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The complexity of election problems with group-separable preferences

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Abstract

We analyze the complexity of several \({{\mathrm {NP}}}\)-hard election-related problems under the assumptions that the voters have group-separable preferences. We show that under this assumption our problems typically remain \({{\mathrm {NP}}}\)-hard, but we provide more efficient algorithms if additionally the clone decomposition tree is of moderate height. We also show a polynomial-time algorithm for sampling group-separable elections uniformly at random.

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Notes

  1. Yet, we mention that occasionally tie-breaking may affect the complexity of election problems; see, e.g., the works of Obraztsova et al. [33, 34].

  2. We mention that there is also a notion of single-crossing width of an election, which can be quite useful algorithmically [16]. The notions of single-peaked width and single-crossing width are particularly relevant to our work because, as in the case of group-separable elections, they rely on clone decomposition trees. However, their use of these trees is quite different than the one for group-separable elections.

  3. CCAC stands for constructive control by adding candidates.

  4. CCAV stands for constructive control by adding voters.

  5. The fact that every election has an (in essence) unique clone decomposition tree is included as part of a discussion in the work of Elkind et al. [19], but it is not stated as a theorem there. Karpov [28] provides a formal statement of this result, together with a proof. Based on that, he shows his characterization of group-separable elections.

  6. Note that the core path in this tree is, e.g., \(P_1, \ldots , P_{m-1},c_{m}\).

  7. The reader may wonder why we speak of a size-k committee here, and not of an up-to-size-k committee. The reason is that we minimize the dissatisfaction, so if there is a committee of size smaller than k with a given dissatisfaction, then there certainly is a committee of size exactly k with at least as small a dissatisfaction.

  8. We note that during this greedy process we might end up deleting all the candidates in some set \(L_i\) or \(R_i\). If this happens then we terminate our algorithm for this strategy of increasing the score of p because, effectively, we have moved into another one, which we will consider later (or, which we have considered already).

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Acknowledgements

This work was supported by the MOE (Singapore) under Tier-1 Grant 2018-T1-001-118 and by the NTU SUG M4081985 Grant. Alexander Karpov was supported by the Basic Research Program of the National Research University Higher School of Economics.

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Authors and Affiliations

  1. AGH University, Kraków, Poland

    Piotr Faliszewski

  2. HSE Universtiy, Moscow, Russia

    Alexander Karpov

  3. Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    Alexander Karpov

  4. Nanyang Technological University, Singapore, Singapore

    Svetlana Obraztsova

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  1. Piotr Faliszewski
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  2. Alexander Karpov
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  3. Svetlana Obraztsova
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Correspondence to Piotr Faliszewski.

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Faliszewski, P., Karpov, A. & Obraztsova, S. The complexity of election problems with group-separable preferences. Auton Agent Multi-Agent Syst 36, 18 (2022). https://doi.org/10.1007/s10458-022-09549-7

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