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2013 An Improved Exact Algorithm for Least-Squares Unidimensional Scaling
Gintaras Palubeckis
J. Appl. Math. 2013: 1-15 (2013). DOI: 10.1155/2013/890589


Given n objects and an n×n 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 36×36 Morse code dissimilarity matrix.


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Gintaras Palubeckis. "An Improved Exact Algorithm for Least-Squares Unidimensional Scaling." J. Appl. Math. 2013 1 - 15, 2013.


Published: 2013
First available in Project Euclid: 14 March 2014

zbMATH: 1266.65072
MathSciNet: MR3049451
Digital Object Identifier: 10.1155/2013/890589

Rights: Copyright © 2013 Hindawi

Vol.2013 • 2013
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