Nonnegative integral subset representations of integer sets
Information Processing Letters
Approximation algorithms for minimizing segments in radiation therapy
Information Processing Letters
Minimum Cardinality Matrix Decomposition into Consecutive-Ones Matrices: CP and IP Approaches
CPAIOR '07 Proceedings of the 4th international conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
Computers and Operations Research
The sum-of-increments constraint in the consecutive-ones matrix decomposition problem
Proceedings of the 2009 ACM symposium on Applied Computing
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
Decomposition of integer matrices and multileaf collimator sequencing
Discrete Applied Mathematics
A new algorithm for optimal multileaf collimator field segmentation
Discrete Applied Mathematics
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
Generalized geometric approaches for leaf sequencing problems in radiation therapy
ISAAC'04 Proceedings of the 15th international conference on Algorithms and Computation
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
On explaining integer vectors by few homogenous segments
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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The segment minimization problem consists of finding a smallest set of binary matrices (segments), where non-zero values in each row of each matrix are consecutive, each matrix is assigned a positive integer weight (a segment-value), and the weighted sum of the matrices corresponds to the given input intensity matrix. This problem has direct applications in intensity-modulated radiation therapy, an effective form of cancer treatment. We study here the special case when the largest value H in the intensity matrix is small. We show that for an intensity matrix with one row, this problem is fixed-parameter tractable (FPT) in H; our algorithm obtains a significant asymptotic speedup over the previous best FPT algorithm. We also show how to solve the full-matrix problem faster than all previously known algorithms. Finally, we address a closely related problem that deals with minimizing the number of segments subject to a minimum beam-on time, defined as the sum of the segment-values, and again improve the running time of previous algorithms. Our algorithms have running time O(mn) in the case that the matrix has only entries in {0,1,2}.