Extreme Point Solutions for Infinite Network Flow Problems

We study capacitated network flow problems with supplies and demands defined on a countably infinite collection of nodes having finite degree. This class of network flow models includes, for example, all infinite horizon deterministic dynamic programs with finite action sets since these are equivalent to the problem of finding a shortest infinite path in an infinite directed network. We derive necessary and sufficient conditions for flows to be extreme points of the set of feasible flows. Under a regularity condition met by all such problems with integer data, we show that a feasible solution is an extreme point if and only if it contains no finite or infinite cycles of free arcs (an arc is free if its flow is strictly between its upper and lower bounds). We employ this characterization to establish the integrality of extreme point flows whenever demands and supplies and arc capacities are integer valued. We illustrate our results with an application to an infinite horizon economic lot-sizing problem.