Monotonicity and conformality in multicommodity network‐flow problems

The objective of this paper is to develop a monotonicity theory for the important class of minimum convex‐cost parametric multicommodity network‐flow problems defined over directed graphs. The results allow us to determine when it is possible to predict, without numerical computations, the direction of change of optimal multicommodity flows resulting from changes in arc‐commodity parameters. In particular, we provide necessary and sufficient conditions that for every cost function satisfying some convexity and submodularity assumptions there always exists an optimal multicommodity flow for which the flow of a commodity in a given arc a is nondecreasing (resp., nonincreasing) in the parameter of a distinct commodity in arc b. These conditions are that either (1) there are only two commodities and the underlying undirected graph is series‐parallel or (2) there are three or more commodities and the graph is 2‐isomorphic to a suspension graph. A characterization of the precise pairs of arcs for which the above monotonicity result holds is also provided.