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

Abstract

Given a finite set X and a collection Π=(R1,R2,…,Rv) 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(Ri,R) between R and the binary relations Ri of Π (1≤i≤v), which measures the number of disagreements between Ri and R. Usually, the considered remoteness between Π and a relation R is the remoteness ρ1(Π,R) given by the sum of the distances δ(Ri,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 ρ1 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(Ri,R)’s for any integer p, the maximum of the δ(Ri,R)’s, the minimum of the δ(Ri,R)’s, and different kinds of means of the δ(Ri,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.