A note on improving the performance of approximation algorithms for radiation therapy

Abstract

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 non-zeroes 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 constant-factor 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 real-world clinical data. Our algorithms are superior for 76% of instances and we argue for their utility alongside the heuristic approaches used in practice.

Faster Optimal Algorithms for Segment Minimization with Small Maximal Value

“…This result also has implications for the approximation algorithms in [6] where it can be employed as a subroutine to improve results in practice.…”

“…Work by Collins et al [10] shows that the single-column version of the problem is NP-complete and provides some nontrivial lower bounds given certain constraints. Work by Luan et al [16] gives two approximation algorithms for the full Ñ ¢ Ò segmentation problem, and Biedl et al [6] extend this work to achieve better approximation algorithms that result in performance improvements.…”

On Explaining Integer Vectors by Few Homogenous Segments

“…VE occurs in the database context and VE + occurs in the radiation therapy context. Motivated by previous work providing polynomial-time solvable special cases [1,4], polynomial-time approximation [5,19] and fixed-parameter tractability results [6,8] (approximation and fixed-parameter algorithms both exploit problem-specific structural parameters), we head on a systematic parameterized and multivariate complexity analysis [13,21] of both problems; see Table 1 for a survey of parameterized complexity results.…”

“…Concerning computational complexity, VE + is known to be strongly NP-hard [3] and APX-hard [4]. A significant amount of work has been done to achieve approximation algorithms for minimizing the number of segments which improve on the straightforward factor of two [4] (also see Biedl et al [5]). Improving a previous fixed-parameter tractability result for the parameter "maximum value γ of a vector entry" by Cambazard et al [8], Biedl et al [6] developed a fixed-parameter algorithm solving VE + in polynomial time when γ = O((log n) 2 ) with n being the number of entries in the input vector.…”

“…Improving a previous fixed-parameter tractability result for the parameter "maximum value γ of a vector entry" by Cambazard et al [8], Biedl et al [6] developed a fixed-parameter algorithm solving VE + in polynomial time when γ = O((log n) 2 ) with n being the number of entries in the input vector. Moreover, the parameter "maximum difference between two consecutive vector entries" has been exploited for developing polynomial-time approximation algorithms [5,19]. Finally, we remark that most of the previous studies also looked at the two-dimensional ("matrix") case, whereas we focus on the one-dimensional ("vector") case.…”