Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Decomposition of integer matrices and multileaf collimator sequencing
Discrete Applied Mathematics
A new algorithm for optimal multileaf collimator field segmentation
Discrete Applied Mathematics
Minimizing setup and beam-on times in radiation therapy
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Realization of intensity modulated radiation fields using multileaf collimators
General Theory of Information Transfer and Combinatorics
A Shortest Path-Based Approach to the Multileaf Collimator Sequencing Problem
CPAIOR '09 Proceedings of the 6th International Conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
A note on improving the performance of approximation algorithms for radiation therapy
Information Processing Letters
Faster optimal algorithms for segment minimization with small maximal value
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
A shortest path-based approach to the multileaf collimator sequencing problem
Discrete Applied Mathematics
Hybrid methods for the multileaf collimator sequencing problem
CPAIOR'10 Proceedings of the 7th international conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
Faster optimal algorithms for segment minimization with small maximal value
Discrete Applied Mathematics
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The use of multileaf collimators (MLCs) is a modern way to realize intensity modulated fields in radiotherapy. An important step in the treatment planning is the shape matrix decomposition: the desired fluence distribution, given by an integer matrix, has to be decomposed into a small number shape matrices, i.e. (0,1)-matrices corresponding to the field shapes that can be delivered by the MLC used. The two main objectives are to minimize the total irradiation time, and the number of shape matrices. Assuming that the entries of the fluence matrix are bounded by a constant, we prove that a shape matrix decomposition with minimal number of shape matrices under the condition that the total irradiation time is minimal, can be determined in time polynomial in the matrix dimensions. The results of our algorithm are compared with Engel's [K. Engel, A new algorithm for optimal multileaf collimator field segmentation, Discrete Appl. Math. 152 (1-3) (2005) 35-51.] heuristic for the reduction of the number of shape matrices.