Decomposition of integer matrices and multileaf collimator sequencing

Authors:
Davaatseren Baatar;Horst W. Hamacher;Matthias Ehrgott;Gerhard J. Woeginger
Affiliations:
Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany;Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany;Department of Engineering Science, The University of Auckland, Auckland, New Zealand;Department of Mathematics and Computer Science, Eindhoven University of Technology, MB Eindhoven, The Netherlands
Venue:
Discrete Applied Mathematics
Year:
2005

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Abstract

In this paper, we consider the problem of decomposing an integer matrix into a weighted sum of binary matrices that have the strict consecutive ones property. This problem is motivated by an application in cancer radiotherapy planning, namely the sequencing of multileaf collimators to realize a given intensity matrix. In addition, we also mention another application in the design of public transportation. We are interested in two versions of the problem, minimizing the sum of the coefficients in the decomposition (decomposition time) and minimizing the number of matrices used in the decomposition (decomposition cardinality). We present polynomial time algorithms for unconstrained and constrained versions of the decomposition time problem and prove that the (unconstrained) decomposition cardinality problem is strongly NP-hard. For the decomposition cardinality problem, some polynomially solvable special cases are considered and heuristics are proposed for the general case.