I n this paper, we introduce an exact method based on constraint programming ideas for a combinatorial optimization problem that arises from the treatment planning of intensity-modulated radiotherapy-the minimum cardinality problem (MCP). The MCP is to find a decomposition of a given integer matrix into a weighted sum of binary matrices with consecutive ones, such that the number of such binary matrices is minimised. We compare our method with two recent exact methods for the same problem and a recent exact method for a special case of the problem. Numerical results are presented that indicate that our method is computationally more efficient than the three existing methods.

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We propose a parallel algorithm for computing exact solutions to the problem of minimizing the number of multileaf collimator apertures needed in stepand-shoot intensity modulated radiotherapy. These problems are very challenging particularly as the problem size increases. Here, we investigate how advanced parallel computing methods can be applied to these problems with a focus on the issues that are peculiar to parallel search algorithms and do not arise in their serial counterparts. A previous paper by the authors presented the MU-RD method for solving such problems using a serial constraint programming based search method. This method is being used as the starting point for a parallel implementation. The key challenges in creating a parallel implementation are ensuring that the CPUs are not starved of work and avoiding unnecessary computation due to the rearrangement of the search order in the parallel version. We show that efficient parallel optimisation is possible by dynamically changing the way work is split with potentially multiple tree search processes as well as parallel search of nodes. A weakly sorted queueing system is used to ensure appropriate prioritisation of tasks. Numerical results are presented to demonstrate the effectiveness of our algorithms in scaling from 8 to 64 CPUs.