Seriation in the presence of errors: NP-hardness of l∞-fitting Robinson structures to dissimilarity matrices

In this paper, we establish that the following fitting problem is NP-hard: given a finite set X and a dissimilarity measure d on X (d is a symmetric function d from X to the nonnegative real numbers and vanishing on the diagonal), we wish to find a Robinsonian dissimilarity dR on X minimizing the l∞-error ||d− dR||∞ = maxx,y∈X{|d(x, y)− dR(x, y)|} between d and dR. Recall that a dissimilarity dR on X is called monotone (or Robinsonian) if there exists a total order ≺ on X such that x ≺ z ≺ y implies that d(x, y) ≥ max{d(x, z), d(z, y)}. The Robinsonian dissimilarities appear in seriation and clustering problems, in sparse matrix ordering and DNA sequencing.