Polynomial time algorithms for solving the multicommodity flow problems on two types of directed cycles

This paper considers the two kinds of integer multicommodity flow problems, a feasibility problem and a maximization problem, on two types of directed cycles, a unidirectional and a bidirectional cycle. Both multicommodity flow problems on an undirected cycle have been dealt with by many researchers and it is known that each problems can be solved by a polynomial time algorithm. How-ever, we don't find any result on the directed cycles. Here we show that we can also solve both problems for a unidirectional and a bidirectional cycle in polynomial time.