Seriation in the Presence of Errors: NP-Hardness of l ∞ -Fitting Robinson Structures to Dissimilarity Matrices

  • Authors:

  • Victor Chepoi;Bernard Fichet;Morgan Seston

  • Affiliations:

  • Université de la Méditerranée, Laboratoire d’Informatique Fondamentale de Marseille, Faculté des Sciences de Luminy, F-13288, Marseille Cedex 9, France;Université de la Méditerranée, Laboratoire d’Informatique Fondamentale de Marseille, Faculté des Sciences de Luminy, F-13288, Marseille Cedex 9, France;Université de la Méditerranée, Laboratoire d’Informatique Fondamentale de Marseille, Faculté des Sciences de Luminy, F-13288, Marseille Cedex 9, France

  • Venue:
  • Journal of Classification
  • Year:
  • 2009

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Abstract

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 from X2 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 − d R||∞ = 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.