Fitting points on the real line and its application to RH mapping

Abstract

A natural problem is that of, given an n× n symmetric matrix D, finding an arrangement of n points on the real line such that the so obtained distances agree as well as possible with the by D specified distances. We refer to the variation in which the difference in distance is measured in maximum norm as the Matrix-To-Line problem. The Matrix-To-Line problem has previously been shown to be NP-complete [J.B. Saxe, 17th Allerton Conference in Communication, Control, and Computing, 1979, pp. 480–489]. We show that it can be approximated within 2, but unless P=NP not within 7/5− δ for any δ>0. We also show a tight lower bound under a stronger assumption. We show that the Matrix-To-Line problem cannot be approximated within 2− δ unless 3-colorable graphs can be colored with ⌈4/ δ⌉ colors in polynomial time. Currently, the best polynomial time algorithm colors a 3-colorable graph with O(n 3/14) colors [A. Blum, D. Karger, Inform. Process. Lett. 61 (1), (1997), 49–53]. We apply our Matrix-To-Line algorithm to a problem in computational biology, namely, the Radiation Hybrid (RH) problem. That is, the algorithmic part of a physical mapping method called RH mapping. This gives us the first algorithm with a guaranteed convergence for the general RH problem.