A network flow algorithm to minimize beam-on time for unconstrained multileaf collimator problems in cancer radiation therapy

In this article, we study the modulation of intensity matrices arising in cancer radiation therapy using multileaf collimators. This problem can be formulated by decomposing a given m × n integer matrix into a positive linear combination of l0, 1r matrices with the strict consecutive 1's property in rows. We consider a special case in which no technical constraints have to be taken into account. In this situation, the rows of the intensity matrix are independent of each other and the problem is equivalent to decomposing m intensity rows—independent of each other—into positive linear combinations of l0, 1r rows with the consecutive 1's property. We demonstrate that this problem can be transformed into a minimum cost flow problem in a directed network that has the following special structures: l1r the network is acyclic; l2r it is a complete graph lthat is, there is an arc li, jr whenever i < jr; l3r each arc cost is 1; and l4r each arc is uncapacitated lthat is, it has infinite capacityr. We show that using this special structure, the minimum cost flow problem can be solved in Olnr time. Because we need to solve m such problems, the total running time of our algorithm is Olnmr, which is an optimal algorithm to decompose a given m × n integer matrix into a positive linear combination of l0, 1r matrices. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 45l1r, 36–41 2005