Minimizing maximum flows in linear graphs

We define a linear graph to be a connected acyclic graph each of whose nodes is of degree one or two. We consider a flow problem in linear graphs in which a commodity flows from source nodes to sink nodes. Each source node has a specified value denoting an amount of a commodity to be disposed of and each sink node has a specified capacity denoting the maximum amount of the commodity it can absorb; edges are capable of carrying an arbitrary quantity of the commodity. A solution is a set of flows which transports the commodity from all source nodes to sink nodes without overfilling any sink. A solution is defined to be optimal if it is minimax, that is, the largest flow along any edge is as small as possible. We describe 0(n 2 ) and 0(n) algorithms for finding optimal solutions.