Extreme point characterizations for infinite network flow problems

  • Authors:

  • H. Edwin Romeijn;Dushyant Sharma;Robert L. Smith

  • Affiliations:

  • Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, P.O. Box 116595, Gainesville, Florida 32611-6595;Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, Michigan 48109-2117;Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, Michigan 48109-2117

  • Venue:
  • Networks
  • Year:
  • 2006

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Abstract

We study capacitated network flow problems with 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, because 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 neither a cycle nor a doubly-infinite path consisting 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 node demands and arc capacities are integer valued. We illustrate our results with an application to an infinite horizon economic lot-sizing problem. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(4), 209–222 2006