An Approximation Algorithm for l\infty-Fitting Robinson Structures to Distances

TL;DR

This paper introduces a factor 16 approximation algorithm for the NP-hard problem of fitting Robinsonian distances to a given distance matrix, which is relevant in seriation and classification tasks.

Contribution

The paper presents the first approximation algorithm with a factor of 16 for the NP-hard Robinsonian distance fitting problem.

Findings

Provides a factor 16 approximation algorithm for the problem.

Addresses NP-hardness of fitting Robinsonian distances.

Applicable to seriation and classification problems.

Abstract

In this paper, we present a factor 16 approximation algorithm for the following NP-hard distance fitting problem: given a finite set X and a distance d on X, find a Robinsonian distance dR on X minimizing the l\infty-error ||d - dR||\infty = maxx,y\epsilonX {|d(x, y) - dR(x, y)|}. A distance dR on a finite set X is Robinsonian if its matrix can be symmetrically permuted so that its elements do not decrease when moving away from the main diagonal along any row or column. Robinsonian distances generalize ultrametrics, line distances and occur in the seriation problems and in classification.