Fitting points on the real line and its application to RH mapping
2003 (English)In: Journal of Algorithms, ISSN 0196-6774, E-ISSN 1090-2678, Vol. 49, no 1, 42-62 p.Article in journal (Refereed) Published
A natural problem is that of, given an n x 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 - delta for any delta > 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 - delta unless 3-colorable graphs can be colored with [4/delta] colors in polynomial time. Currently, the best polynomial time algorithm colors a 3-colorable graph with (O) over tilde (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.
Place, publisher, year, edition, pages
2003. Vol. 49, no 1, 42-62 p.
line fitting, RH mapping, approximation, lower bounds, graphs
IdentifiersURN: urn:nbn:se:kth:diva-22909DOI: 10.1016/s0196-6774(03)00083-xISI: 000186142500004OAI: oai:DiVA.org:kth-22909DiVA: diva2:341607
QC 201005252010-08-102010-08-10Bibliographically approved