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Complexity results for extensions of median orders to different types of remoteness

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  • Olivier Hudry

Abstract

Given a finite set X and a collection Π=(R 1 ,R 2 ,…,R v ) of v binary relations defined on X and given a remoteness ρ, a relation R is said to be a central relation of Π with respect to ρ if it minimizes the remoteness ρ(Π,R) from Π. The remoteness ρ is based on the symmetric difference distance δ(R i ,R) between R and the binary relations R i of Π (1≤i≤v), which measures the number of disagreements between R i and R. Usually, the considered remoteness between Π and a relation R is the remoteness ρ 1 (Π,R) given by the sum of the distances δ(R i ,R) over i, and thus measures the total number of disagreements between Π and R or, divided by v, provides the (arithmetical) mean number of disagreements between Π and R. The computation of a central relation with respect to ρ 1 is often an NP-hard problem when the central relation is required to fulfill structural properties like transitivity. In this paper, we investigate other types of remoteness ρ, for instance the sum of the pth power of the δ(R i ,R)’s for any integer p, the maximum of the δ(R i ,R)’s, the minimum of the δ(R i ,R)’s, and different kinds of means of the δ(R i ,R)’s, or their weighted versions. We show that for many definitions of the remoteness, including the previous ones, the computation of a central relation with respect to ρ remains an NP-hard problem, even when the number v of relations is given, for any value of v greater than or equal to 1. Copyright Springer Science+Business Media New York 2015

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  • Olivier Hudry, 2015. "Complexity results for extensions of median orders to different types of remoteness," Annals of Operations Research, Springer, vol. 225(1), pages 111-123, February.
    • Handle: RePEc:spr:annopr:v:225:y:2015:i:1:p:111-123:10.1007/s10479-013-1342-3
    DOI: 10.1007/s10479-013-1342-3
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    References listed on IDEAS

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    1. Hudry, Olivier, 2009. "A survey on the complexity of tournament solutions," Mathematical Social Sciences, Elsevier, vol. 57(3), pages 292-303, May.
    2. Bernard Monjardet & Jean-Pierre Barthélemy & Olivier Hudry & Bruno Leclerc, 2009. "Metric and latticial medians," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00408174, HAL.
    3. Olivier Hudry & Bernard Monjardet, 2010. "Consensus theories: an oriented survey," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00504974, HAL.
    4. Pierre Barthelemy, Jean & Monjardet, Bernard, 1981. "The median procedure in cluster analysis and social choice theory," Mathematical Social Sciences, Elsevier, vol. 1(3), pages 235-267, May.
    5. Hudry, Olivier, 2012. "On the computation of median linear orders, of median complete preorders and of median weak orders," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 2-10.
    6. Olivier Hudry, 2008. "NP-hardness results for the aggregation of linear orders into median orders," Annals of Operations Research, Springer, vol. 163(1), pages 63-88, October.
    7. Denis Bouyssou & Thierry Marchant & Marc Pirlot & Alexis Tsoukiàs & Philippe Vincke, 2006. "Evaluation and Decision Models with Multiple Criteria," International Series in Operations Research and Management Science, Springer, number 978-0-387-31099-2, December.
    8. Hudry, Olivier, 2010. "On the complexity of Slater's problems," European Journal of Operational Research, Elsevier, vol. 203(1), pages 216-221, May.
    9. Irène Charon & Olivier Hudry, 2010. "An updated survey on the linear ordering problem for weighted or unweighted tournaments," Annals of Operations Research, Springer, vol. 175(1), pages 107-158, March.
    10. Nathalie Caspard & Bruno Leclerc & Bernard Monjardet, 2012. "Finite Ordered Sets Concepts, Results and Uses," Post-Print halshs-00800193, HAL.
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