An Integer Linear Programming Formulation for the Minimum Cardinality Segmentation Problem

In this article, we investigate the Minimum Cardinality Segmentation Problem (MCSP), an NP-hard combinatorial optimization problem arising in intensity-modulated radiation therapy. The problem consists in decomposing a given nonnegative integer matrix into a nonnegative integer linear combination of a minimum cardinality set of binary matrices satisfying the consecutive ones property. We show how to transform the MCSP into a combinatorial optimization problem on a weighted directed network and we exploit this result to develop an integer linear programming formulation to exactly solve it. Computational experiments show that the lower bounds obtained by the linear relaxation of the considered formulation improve upon those currently described in the literature and suggest, at the same time, new directions for the development of future exact solution approaches to the problem.