A Shortest Path-Based Approach to the Multileaf Collimator Sequencing Problem

The multileaf collimator sequencing problem is an important component in effective cancer treatment delivery. The problem can be formulated as finding a decomposition of an integer matrix into a weighted sequence of binary matrices whose rows satisfy a consecutive ones property. Minimising the cardinality of the decomposition is an important objective and has been shown to be strongly NP-Hard, even for a matrix restricted to a single row. We show that in this latter case it can be solved efficiently as a shortest path problem, giving a simple proof that the one line problem is fixed-parameter tractable in the maximum intensity. This result was obtained recently by [9] with a complex construction. We develop new linear and constraint programming models exploiting this idea. Our approaches significantly improve the best known for the problem, bringing real-world sized problem instances within reach of complete methods.