Approximation and Hardness of Shift-Bribery

Faliszewski, Piotr; Manurangsi, Pasin; Sornat, Krzysztof

doi:10.1609/aaai.v33i01.33011901

Cited by 9 publications

(7 citation statements)

References 19 publications

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“…Related areas such as Fixed Parameter Tractability (FPT) 7 and hardness of approximation 8 have already gained significant traction in computational social choice. See, e.g., the following surveys on FPT results and further challenges in computational social choice [10,14,22], and, e.g., the following papers on hardness of approximation in bribery problems [21], multiwinner elections [3,4,17,43] and a survey on fair-division with indivisible goods [31].…”

Section: Our Contributionmentioning

confidence: 99%

Fine-Grained Complexity and Algorithms for the Schulze Voting Method

Sornat

Williams

2021

Proceedings of the 22nd ACM Conference on Economics and Computation

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We study computational aspects of a well-known single-winner voting rule called the Schulze method [Schulze, 2003] which is used broadly in practice. In this method the voters give (weak) ordinal preference ballots which are used to define the weighted majority graph of direct comparisons between pairs of candidates. The choice of the winner comes from indirect comparisons in the graph, and more specifically from considering directed paths instead of direct comparisons between candidates.When the input is the weighted majority graph, to our knowledge, the fastest algorithm for computing all winners in the Schulze method uses a folklore reduction to the All-Pairs Bottleneck Paths (APBP) problem and runs in O(m 2.69 ) time, where m is the number of candidates. It is an interesting open question whether this can be improved. Our first result is a combinatorial algorithm with a nearly quadratic running time for computing all winners. This running time is essentially optimal as it is nearly linear in the size of the weighted majority graph.If the input to the Schulze winners problem is not the weighted majority graph but the preference profile, then constructing the weighted majority graph is a bottleneck that increases the running time significantly; in the special case when there are m candidates and n = O(m) voters, the running time is O(m 2.69 ), or O(m 2.5 ) if there is a nearly-linear time algorithm for multiplying dense square matrices.To address this bottleneck, we prove a formal equivalence between the well-studied Dominance Product problem and the problem of computing the weighted majority graph. As the Dominance Product problem is believed to require at least time r 2.5−o(1) on r × r matrices, our equivalence implies that constructing the weighted majority graph in O(m 2.499 ) time for m candidates and n = O(m) voters would imply a breakthrough in the study of "intermediate" problems [Lincoln et al., 2020] in fine-grained complexity. We prove a similar connection between the so called Dominating Pairs problem and the problem of verifying whether a given candidate is a winner.Our paper is the first to bring fine-grained complexity into the field of computational social choice. Previous approaches say nothing about lower bounds for problems that already have polynomial time algorithms. By bringing fine-grained complexity into the picture we can identify voting protocols that are unlikely to be practical for large numbers of candidates and/or voters, as their complexity is likely, say at least cubic. CCS Concepts: • Theory of computation → Algorithmic game theory and mechanism design; Graph algorithms analysis; • Applied computing → Voting / election technologies.

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Section: Our Contributionmentioning

confidence: 99%

Fine-Grained Complexity and Algorithms for the Schulze Voting Method

Sornat

Williams

2021

Proceedings of the 22nd ACM Conference on Economics and Computation

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“…Related areas such as Fixed Parameter Tractability (FPT) 7 and hardness of approximation 8 have already gained significant traction in computational social choice. See, e.g., the following surveys on FPT results and further challenges in computational social choice [BCF + 14, FN16, DS17], and, e.g., the following papers on hardness of approximation in bribery problems [FMS19], multiwinner elections [SFL16, BFGG20, DMMS20, BFF20] and a survey on fair-division with indivisible goods [Mar17].…”

Section: Our Contributionmentioning

confidence: 99%

Fine-Grained Complexity and Algorithms for the Schulze Voting Method

Sornat

Williams

2021

Preprint

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We study computational aspects of a well-known single-winner voting rule called the Schulze method [Schulze, 2003] which is used broadly in practice. In this method the voters give (weak) ordinal preference ballots which are used to define the weighted majority graph of direct comparisons between pairs of candidates. The choice of the winner comes from indirect comparisons in the graph, and more specifically from considering directed paths instead of direct comparisons between candidates.When the input is the weighted majority graph, to our knowledge, the fastest algorithm for computing all possible winners in the Schulze method uses a folklore reduction to the All-Pairs Bottleneck Paths (APBP) problem and runs in O(m 2.69 ) time, where m is the number of candidates. It is an interesting open question whether this can be improved. Our first result is a combinatorial algorithm with a nearly quadratic running time for computing all possible winners. This running time is essentially optimal as it is nearly linear in the size of the weighted majority graph.If the input to the possible winners problem is not the weighted majority graph but the preference profile, then constructing the weighted majority graph is a bottleneck that increases the running time significantly; in the special case when there are O(m) voters and candidates, the running time becomes O(m 2.69 ), or O(m 2.5 ) if there is a nearly-linear time algorithm for multiplying dense square matrices.To address this bottleneck, we prove a formal equivalence between the well-studied Dominance Product problem and the problem of computing the weighted majority graph. As the Dominance Product problem is believed to require at least time r 2.5−o(1) on r × r matrices, our equivalence implies that constructing the weighted majority graph in O(m 2.499 ) time for O(m) voters and candidates would imply a breakthrough in the study of "intermediate" problems [Lincoln et al., 2020] in fine-grained complexity. We prove a similar connection between the so called Dominating Pairs problem and the problem of verifying whether a given candidate is a possible winner.Our paper is the first to bring fine-grained complexity into the field of computational social choice. Previous approaches say nothing about problems that already have polynomial time algorithms. By bringing fine-grained complexity into the picture we can identify voting protocols that are unlikely to be practical for large numbers of candidates and/or voters, as their complexity is likely, say at least cubic.

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“…SWAP-and SHIFT-BRIBERY were introduced by Elkind et al [16]. Various authors studied these problems for different voting rules (see, e.g., the works of Maushagen et al [27] and Zhou and Guo [37] regarding iterative elections), sought approximation algorithms [15,18], established parameterized complexity results [14,8,23], considered restricted preference domains [17], and extended the problem in various ways [6,21,4,36]. The idea of using SWAP-BRIBERY to measure the robustness of election results is due to Shiryaev et al [30], but is also closely related to computing the margin of victory [25,11,35,10]; recently it was also applied to committee elections [7].…”

Section: Related Workmentioning

confidence: 99%

On the Robustness of Winners: Counting Briberies in Elections

Boehmer,

Bredereck,

Faliszewski

et al. 2020

Preprint

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Approximation and Hardness of Shift-Bribery

Cited by 9 publications

References 19 publications

Fine-Grained Complexity and Algorithms for the Schulze Voting Method

Fine-Grained Complexity and Algorithms for the Schulze Voting Method

Fine-Grained Complexity and Algorithms for the Schulze Voting Method

On the Robustness of Winners: Counting Briberies in Elections

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