SIAM Journal on Discrete Mathematics
The hardness of approximation: gap location
Computational Complexity
Page replacement for general caching problems
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
A unifying look at data structures
Communications of the ACM
A unified approach to approximating resource allocation and scheduling
Journal of the ACM (JACM)
Scaling and related techniques for geometry problems
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Mountain reduction, block matching, and applications in intensity-modulated radiation therapy
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
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
Algorithmics in intensity-modulated radiation therapy
Algorithms and theory of computation handbook
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
Faster optimal algorithms for segment minimization with small maximal value
Discrete Applied Mathematics
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Radiation therapy is one of the commonly used cancer therapies. The radiation treatment poses a tuning problem: it needs to be effective enough to destroy the tumor, but it should maintain the functionality of the organs close to the tumor. Towards this goal the design of a radiation treatment has to be customized for each patient. Part of this design are intensity matrices that define the radiation dosage in a discretization of the beam head. To minimize the treatment time of a patient the beam-on time and the setup time need to be minimized. For a given row of the intensity matrix, the minimum beam-on time is equivalent to the minimum number of binary vectors with the consecutive “1”s property that sum to this row, and the minimum setup time is equivalent to the minimum number of distinct vectors in a set of binary vectors with the consecutive “1”s property that sum to this row. We give a simple linear time algorithm to compute the minimum beam-on time. We prove that the minimum setup time problem is APX-hard and give approximation algorithms for it using a duality property. For the general case, we give a $\frac {24}{13}$ approximation algorithm. For unimodal rows, we give a $\frac 97$ approximation algorithm. We also consider other variants for which better approximation ratios exist.