Faster optimal algorithms for segment minimization with small maximal value

Abstract

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}.