ZIB PaperWeb

SC 92-04

Yoshiko Wakabayashi Medians of Binary Relations: Computational Complexity.

Abstract: Let $\Re$ be the set of all binary relations on a finite set $N$ and $d$ be the symmetric difference distance defined on $\Re$. For a given profile $\Pi = (R_1,...,R_m) \in R^m$, a relation $R* \in \Re$ that minimizes the function $\sum^m_{k=1} d(R_k,R)$ is called a median relation of $\Pi$. A number of problems occuring 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 $\Pi$ represents collected data (preferences, similarities, games) and the objective is that of finding a median relation of $\Pi$ 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: reflexitivity, 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 $\Pi$ has a fixed number of binary relations.