Fixed-Parameter Tractability and Improved Approximations for Segment Minimization

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 non-zeroes in each row are consecutive. This has direct applications in intensity-modulated radiation therapy, an effective form of cancer treatment. We show here that for the case of an 1 × n-matrix, this problem is fixed-parameter tractable in the largest value of the intensity matrix. We use this to develop approximation algorithms for matrices with arbitrarily many rows. One of these improves the approximation factor from the previous best of log 2 h + 1 to 3/2 · (log 3 h + 1), where h is the largest entry in the intensity matrix; another 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. Experimentation with these algorithms show 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.