Faster optimal algorithms for segmentMinimization with small maximal value*

Abstract

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 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. Here, we obtain an almost-quadratic speedup over the previous best algorithm.