H. Edwin Romeijn, Dushyant Sharma, Robert L. Smith
Extreme point characterizations 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 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 an additional 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 result
to show that the extreme points can be characterized by specifying a basis. Moreover,
we 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.