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Journal of Applied Mathematics
Volume 2013, Article ID 890589, 15 pages
Research Article

An Improved Exact Algorithm for Least-Squares Unidimensional Scaling

Faculty of Informatics, Kaunas University of Technology, Studentu 50-408, 51368 Kaunas, Lithuania

Received 16 October 2012; Accepted 31 March 2013

Academic Editor: Frank Werner

Copyright © 2013 Gintaras Palubeckis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Given n objects and an symmetric dissimilarity matrix D with zero main diagonal and nonnegative off-diagonal entries, the least-squares unidimensional scaling problem asks to find an arrangement of objects along a straight line such that the pairwise distances between them reflect dissimilarities represented by the matrix D. In this paper, we propose an improved branch-and-bound algorithm for solving this problem. The main ingredients of the algorithm include an innovative upper bounding technique relying on the linear assignment model and a dominance test which allows considerably reducing the redundancy in the enumeration process. An initial lower bound for the algorithm is provided by an iterated tabu search heuristic. To enhance the performance of this heuristic we develop an efficient method for exploring the pairwise interchange neighborhood of a solution in the search space. The basic principle and formulas of the method are also used in the implementation of the dominance test. We report computational results for both randomly generated and real-life based problem instances. In particular, we were able to solve to guaranteed optimality the problem defined by a Morse code dissimilarity matrix.