Decomposition of integer matrices and multileaf collimator sequencing

Abstract

In this paper, we consider the problem of decomposing an integer matrix into a weighted sum of binary matrices that have the strict consecutive ones property. This problem is motivated by an application in cancer radiotherapy planning, namely the sequencing of multileaf collimators to realize a given intensity matrix. In addition, we also mention another application in the design of public transportation. We are interested in two versions of the problem, minimizing the sum of the coefficients in the decomposition (decomposition time) and minimizing the number of matrices used in the decomposition (decomposition cardinality). We present polynomial time algorithms for unconstrained and constrained versions of the decomposition time problem and prove that the (unconstrained) decomposition cardinality problem is strongly NP-hard. For the decomposition cardinality problem, some polynomially solvable special cases are considered and heuristics are proposed for the general case.

Integrating column generation in a method to compute a discrete representation of the non-dominated set of multi-objective linear programmes

“…In practice, the intensity pattern is realised by stacking a limited number of shaped radiation fields, each passing through an associated segment. Therefore, after obtaining the intensity pattern as output from solving the FMO problem, it is necessary to solve a so-called segmentation problem, which finds a set of segments that best realise the intensity pattern by, for instance, minimising the total beam-on time required to deliver the intensity pattern or by minimising the required number of segments (see Baatar et al 2005).…”

Decomposition of matrices and static multileaf collimators: a survey

“…A similar idea as in Engel (2005) is used in Baatar et al (2005) for the constrained decomposition cardinality problem. Data from the solution of the DT problem (see Section 2) is used as input for a greedy extraction procedure.…”

On Explaining Integer Vectors by Few Homogenous Segments

“…For a general account on the motivation from radiation therapy refer to the survey by Ehrgott et al [10]. 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]).…”

“…An explanation is a set of segments that sum up to the input vector. For instance, in case of Vector Explanation (VE for short) the vector (4,3,3,4) can be explained by the segments (4,4,4,4) and (0, −1, −1, 0), and in case of Vector Explanation + (VE + for short) it can be explained by (3,3,3,3), (1, 0, 0, 0), and (0, 0, 0, 1).…”