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Algorithms 2015, 8(4), 999-1020; doi:10.3390/a8040999

An Integer Linear Programming Formulation for the Minimum Cardinality Segmentation Problem

1
Louvain School of Management, Center for Operations Research and Econometrics (CORE), Université Catholique de Louvain, Chaussée de Binche 151, bte M1.01.01, 7000 Mons, Belgium
2
Institut Catholique des Hautes Etudes Commerciales (ICHEC), Boulevard Brand Whitlock 2, 1150 Brussels, Belgium
*
Author to whom correspondence should be addressed.
Academic Editors: Giuseppe Lancia and Alberto Policriti
Received: 27 June 2015 / Revised: 23 September 2015 / Accepted: 2 November 2015 / Published: 11 November 2015
(This article belongs to the Special Issue Algorithmic Themes in Bioinformatics)
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Abstract

In this article, we investigate the Minimum Cardinality Segmentation Problem (MCSP), an NP-hard combinatorial optimization problem arising in intensity-modulated radiation therapy. The problem consists in decomposing a given nonnegative integer matrix into a nonnegative integer linear combination of a minimum cardinality set of binary matrices satisfying the consecutive ones property. We show how to transform the MCSP into a combinatorial optimization problem on a weighted directed network and we exploit this result to develop an integer linear programming formulation to exactly solve it. Computational experiments show that the lower bounds obtained by the linear relaxation of the considered formulation improve upon those currently described in the literature and suggest, at the same time, new directions for the development of future exact solution approaches to the problem. View Full-Text
Keywords: matrix decomposition; minimum cardinality segmentation problem; mixed integer linear programming; intensity-modulated radiation therapy; multileaf collimator matrix decomposition; minimum cardinality segmentation problem; mixed integer linear programming; intensity-modulated radiation therapy; multileaf collimator
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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. (CC BY 4.0).

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Catanzaro, D.; Engelbeen, C. An Integer Linear Programming Formulation for the Minimum Cardinality Segmentation Problem. Algorithms 2015, 8, 999-1020.

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