Recognizing Single-Peaked Preferences on an Arbitrary Graph: Complexity and Algorithms

Escoffier, Bruno; Spanjaard, Olivier; Tydrichová, Magdaléna

doi:10.1007/978-3-030-57980-7_19

Cited by 6 publications

(6 citation statements)

References 19 publications

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“…These results were deepened by Bulteau and Chen [7]: they proved that any profile of 3 voters on at most 7 candidates is 2-Euclidean. Finally, Kamiya et al [23], and later Escoffier et al [20], gave a characterization of 2-Euclidean profiles on 4 candidates. Note that there are also some works focusing on metric preferences using l 1 and l ∞ norms [9,20], providing in particular results on the size of Euclidean profiles under these norms, geometrical properties of representations under these norms, and differences between 1 , 2 and ∞ Euclidean profiles.…”

Section: Related Workmentioning

confidence: 99%

“…The nearer a voter is to a candidate, the more this candidate is preferred by the voter. The distance is usually measured by the Euclidean l 2 norm, but the l 1 and l ∞ norms have also been considered [16,29,10,20]. Given a specific domain restriction and a set of preferences (also called preference profile hereafter), a recognition algorithm aims at deciding whether the preferences belong or not to the domain restriction, and if possible also provide a concise certificate of membership or non-membership.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Algorithmic Recognition of 2-Euclidean Preferences

Escoffier,

Spanjaard,

Tydrichová

2023

Frontiers in Artificial Intelligence and Applications

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A set of voters’ preferences on a set of candidates is 2-Euclidean if candidates and voters can be mapped to the plane so that the preferences of each voter decrease with the Euclidean distance between her position and the positions of candidates. Based on geometric properties, we propose a recognition algorithm, that returns either “yes” (together with a planar positioning of candidates and voters) if the preferences are 2-Euclidean, or “no” if it is able to find a concise certificate that they are not, or “unknown” if a time limit is reached. Our algorithm outperforms a quadratically constrained programming solver achieving the same task, both in running times and the percentage of instances it is able to recognize. In the numerical tests conducted on the PrefLib library of preferences, 91.5% (resp. 4.5%) of the available sets of complete strict orders are proven not to be (resp. to be) 2-Euclidean, and the status of only 4.5% of them could not be decided. Furthermore, for instances involving 5 (resp. 6, 7) candidates, we were able to find planar representations that are compatible with 87.4% (resp. 58.1%, 60.1%) of voters’ preferences.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Algorithmic Recognition of 2-Euclidean Preferences

Escoffier,

Spanjaard,

Tydrichová

2023

Frontiers in Artificial Intelligence and Applications

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“…Trick's algorithm only works when voters' preferences are strict; for preferences that may contain ties, more complicated algorithms have been proposed (Trick, 1989a;Conitzer et al, 2004;Tarjan & Yannakakis, 1984;Sheng Bao & Zhang, 2012). Peters and Lackner (2020) give a polynomial-time algorithm for recognizing preferences single-peaked on a circle; very recently, these results have been extended to pseudotrees (Escoffier et al, 2020). On the other hand, a result of Gottlob and Greco (2013) implies that recognizing whether preferences are single-peaked on a graph of treewidth 3 is NP-hard.…”

Section: Related Workmentioning

confidence: 99%

Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results

Peters,

Yu,

Chan

et al. 2020

Preprint

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A preference profile is single-peaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989b) described an efficient algorithm for deciding if a given profile is single-peaked on a tree. We study the complexity of multiwinner elections under several variants of the Chamberlin-Courant rule for preferences single-peaked on trees. We show that in this setting the egalitarian version of this rule admits a polynomial-time winner determination algorithm. For the utilitarian version, we prove that winner determination remains NP-hard, even for the Borda scoring function; however, a winning committee can be found in polynomial time if either the number of leaves or the number of internal vertices of the underlying tree is bounded by a constant. To benefit from these positive results, we need a procedure that can determine whether a given profile is single-peaked on a tree that has additional desirable properties (such as, e.g., a small number of leaves). To address this challenge, we develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is single-peaked. We show how to use this representation to efficiently find the best tree for a given profile for use with our winner determination algorithms: Given a profile, we can efficiently find a tree with the minimum number of leaves, or a tree with the minimum number of internal vertices among trees on which the profile is single-peaked. We then explore the power and limitations of this framework: we develop polynomial-time algorithms to find trees with the smallest maximum degree, diameter, or pathwidth, but show that it is NP-hard to check whether a given profile is single-peaked on a tree that is isomorphic to a given tree, or on a regular tree.

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“…For preferences that may contain ties, more complicated algorithms have been proposed (Trick, 1989a;Conitzer et al, 2004;Tarjan & Yannakakis, 1984;Sheng Bao & Zhang, 2012). Peters and Lackner (2020) give a polynomialtime algorithm for recognizing preferences single-peaked on a circle; very recently, these results have been extended to pseudotrees (Escoffier et al, 2020). On the other hand, a result of Gottlob and Greco (2013) implies that recognizing whether preferences are single-peaked on a graph of treewidth 3 is NP-hard.…”

Section: Related Workmentioning

confidence: 99%

Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results

Peters

Yu²,

Chan³

et al. 2022

jair

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A preference profile is single-peaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989b) described an efficient algorithm for deciding if a given profile is single-peaked on a tree. We study the complexity of multiwinner elections under several variants of the Chamberlin–Courant rule for preferences single-peaked on trees. We show that in this setting the egalitarian version of this rule admits a polynomial-time winner determination algorithm. For the utilitarian version, we prove that winner determination remains NP-hard for the Borda scoring function; indeed, this hardness results extends to a large family of scoring functions. However, a winning committee can be found in polynomial time if either the number of leaves or the number of internal vertices of the underlying tree is bounded by a constant. To benefit from these positive results, we need a procedure that can determine whether a given profile is single-peaked on a tree that has additional desirable properties (such as, e.g., a small number of leaves). To address this challenge, we develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is single-peaked. We show how to use this representation to efficiently find the best tree for a given profile for use with our winner determination algorithms: Given a profile, we can efficiently find a tree with the minimum number of leaves, or a tree with the minimum number of internal vertices among trees on which the profile is single-peaked. We then explore the power and limitations of this framework: we develop polynomial-time algorithms to find trees with the smallest maximum degree, diameter, or pathwidth, but show that it is NP-hard to check whether a given profile is single-peaked on a tree that is isomorphic to a given tree, or on a regular tree.

show abstract

Recognizing Single-Peaked Preferences on an Arbitrary Graph: Complexity and Algorithms

Cited by 6 publications

References 19 publications

Algorithmic Recognition of 2-Euclidean Preferences

Algorithmic Recognition of 2-Euclidean Preferences

Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results

Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results

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