Medians of Binary Relations computational complexity

Abstract: "Let R be the set of all binary relations on a finite set N and d be the symmetric difference distance defined on R. For a given profile II = (Rs, ..., R[subscript m]) [epsilon] R[superscript m], a relation R* [epsilon] R that minimizes the function [formula] is called a median relati...

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Bibliographische Detailangaben
1. Verfasser: Wakabayashi, Yoshiko (VerfasserIn)
Format: UnknownFormat
Sprache:eng
Veröffentlicht: Berlin Konrad-Zuse-Zentrum für Informationstechnik 1992
Schriftenreihe:Konrad-Zuse-Zentrum für Informationstechnik <Berlin>: Preprint SC 1992,4
Schlagworte:
Computational complexity
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Zusammenfassung:Abstract: "Let R be the set of all binary relations on a finite set N and d be the symmetric difference distance defined on R. For a given profile II = (Rs, ..., R[subscript m]) [epsilon] R[superscript m], a relation R* [epsilon] R that minimizes the function [formula] is called a median relation of II. A number of problems occurring [sic] in the social sciences, in qualitative data analysis and in multicriteria decision making can be modelled as problems of finding medians of a profile of binary relations. In these contexts the profile II represents collected data (preferences, similarities, games) and the objective is that of finding a median relation of II with some special feature (representing e.g., consensus of preferences, clustering of similar objects, ranking of teams, etc.)
In this paper we analyse the computational complexity of all such problems in which the median is required to satisfy one or more of the properties: reflexivity, symmetry, antisymmetry, transitivity and completeness. We prove that whenever transitivity is required (except when symmetry and completeness are also simultaneously required) then the corresponding median problem is NP-hard. In some cases we prove that they remain NP-hard when the profile II has a fixed number of binary relations.
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