Constrained Decompositions of Integer Matrices and their Applications to Intensity Modulated Radiation Therapy

We consider combinatorial optimization problems arising in radiation therapy. Given a matrix I with non-negative integer entries, we seek a decomposition of I as a weighted sum of binary matrices having the consecutive ones property, such that the total sum of the coefficients is minimized. The coefficients are restricted to be non-negative integers. Here, we investigate variants of the problem with additional constraints on the matrices used in the decomposition. Constraints appearing in the application include the interleaf motion and interleaf distance constraints. The former constraint was previously studied by Baatar et al. (Discrete Appl. Math., 2005 ) and Kalinowski (Discrete Appl. Math., 2005 ). The latter constraint was independently considered by Kumar et al. (working paper, 2007 ) in the case where coefficients of the decomposition are not restricted to be integers. For both constraints, we prove that finding an optimal decomposition reduces to finding a maximum value potential in an auxiliary network with integer arc lengths and no negative length cycle. This allows us to simplify and unify the previous approaches. Moreover, we give an O(MN + KM) algorithm to solve the problem under the interleaf distance constraint, where M and N respectively denote the number of rows and columns of the matrix I and K is the number of matrices used in the decomposition. We also give an O(MN logM + KM) algorithm for solving the problem under the interleaf motion constraint and hence improve on previous results. Finally, we show the problem can still be solved in O(MN logM +KM) time when both constraints are considered simultaneously.