Faster Optimal Algorithms For Segment Minimization With Small Maximal Value

The segment minimization problem consists of finding the smallest set of integer matrices (segments) that sum to a given intensity matrix, such that each summand has only one non-zero value (the segment-value), and the non-zeroes in each row are consecutive. This has direct applications in intensity-modulated radiation therapy, an effective form of cancer treatment. We study here the special case when the largest value in the intensity matrix is small. We show that for an intensity matrix with one row, this problem is fixed-parameter tractable (FPT) in; 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. Here, we obtain a almost-quadratic speedup over the previous best algorithm

Faster optimal algorithms for segment minimization with small maximal value

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}. © 2012 Elsevier B.V. All rights reserved.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Applications of mathematical network theory

This thesis is a collection of papers on a variety of optimization problems where network structure can be used to obtain efficient algorithms. The considered applications range from the optimization of radiation treatment plkans in cancer therapy to maintenance planning for maximizing the throughput in bulk good supply chains