Minimum Cardinality Matrix Decomposition into Consecutive-Ones Matrices: CP and IP Approaches

  • Authors:

  • Davaatseren Baatar;Natashia Boland;Sebastian Brand;Peter J. Stuckey

  • Affiliations:

  • Department of Mathematics, University of Melbourne, Australia;Department of Mathematics, University of Melbourne, Australia;NICTA Victoria Research Lab, Department of Comp. Sci. and Soft. Eng., University of Melbourne, Australia;NICTA Victoria Research Lab, Department of Comp. Sci. and Soft. Eng., University of Melbourne, Australia

  • Venue:
  • CPAIOR '07 Proceedings of the 4th international conference on Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems
  • Year:
  • 2007

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Abstract

We consider the problem of decomposing an integer matrix into a positively weighted sum of binary matrices that have the consecutive-ones property. This problem is well-known and of practical relevance. It has an important application in cancer radiation therapy treatment planning: the sequencing of multileaf collimators to deliver a given radiation intensity matrix, representing (a component of) the treatment plan.Two criteria characterise the efficacy of a decomposition: the beam-on time(length of time the radiation source is switched on during the treatment), and the cardinality(the number of machine set-ups required to deliver the planned treatment).Minimising the former is known to be easy. However finding a decomposition of minimal cardinality is NP-hard. Progress so far has largely been restricted to heuristic algorithms, mostly using linear programming, integer programming and combinatorial enumerative methods as the solving technologies. We present a novel model, with corresponding constraint programming and integer programming formulations. We compare these computationally with previous formulations, and we show that constraint programming performs very well by comparison.