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A Note on Improving the Performance of Approximation Algorithms for Radiation Therapy
"... The segment minimization problem consists of representing an integer matrix as the sum of the fewest number of integer matrices each of which have the property that the nonzeroes in each row are consecutive. This has direct applications to an effective form of cancer treatment. Using several insigh ..."
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The segment minimization problem consists of representing an integer matrix as the sum of the fewest number of integer matrices each of which have the property that the nonzeroes in each row are consecutive. This has direct applications to an effective form of cancer treatment. Using several insights, we extend previous results to obtain constantfactor improvements in the approximation guarantees. We show that these improvements yield better performance by providing an experimental evaluation of all known approximation algorithms using both synthetic and realworld clinical data. Our algorithms are superior for 76 % of instances and we argue for their utility alongside the heuristic approaches used in practice.
Improved Approximation Algorithms for Segment Minimization in Intensity Modulated Radiation Therapy
, 2009
"... The segment minimization problem consists of finding the smallest set of integer matrices that sum to a given intensity matrix, such that each summand has only one nonzero value, and the nonzeroes in each row are consecutive. This has direct applications in intensitymodulated radiation therapy, a ..."
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The segment minimization problem consists of finding the smallest set of integer matrices that sum to a given intensity matrix, such that each summand has only one nonzero value, and the nonzeroes in each row are consecutive. This has direct applications in intensitymodulated radiation therapy, an effective form of cancer treatment. We develop three approximation algorithms for matrices with arbitrarily many rows. Our first two algorithms improve the approximation factor from the previous best of 1 + log2 h to (roughly) 3/2 · (1 + log3 h) and 11/6 · (1 + log4 h), respectively, where h is the largest entry in the intensity matrix. We illustrate the limitations of the specific approach used to obtain these two algorithms by proving a lower bound of (2b−2) · log b b h + 1 on the approximation guarantee. b Our third algorithm improves the approximation factor from 2 · (log D + 1) to 24/13 · (log D + 1), where D is (roughly) the largest difference between consecutive elements of a row of the intensity matrix. Finally, experimentation with these algorithms shows that they perform well with respect to the optimum and outperform other approximation algorithms on 77 % of the 122 test cases we consider, which include both real world and synthetic data.