Ripples, complements, and substitutes in singly constrained monotropic parametric network flow problems

Abstract

AbstractWe extend the qualitative theory of sensitivity analysis for minimum‐cost flow problems developed by Granot and Veinott to minimum‐cost flow problems with one additional linear constraint. Two natural extensions of the “less dependent on” partial ordering of the arcs are presented. One is decidable in linear time, whereas the other yields more information but is NP‐complete in general. The Ripple Theorem gives upper bounds on the absolute value of optimal‐flow variations as a function of variations in the problem parameters. The theory of substitutes and complements presents necessary and sufficient conditions for optimal‐flow changes to consistently have the same (or the opposite) sign in two given arcs. The Monotonicity Theory links the changes in the value of the parameters to the change in the optimal arc‐flows, and bounds on the rates of changes are discussed. The departure from the pure network structure is shown to have a profound effect on computational issues. Indeed, the complexity of determining substitutes and complements, although linear for the unconstrained (no additional constraint) case, is shown to be NP‐complete in general for the constrained case. However, for all intractable problems, families of cases arise from easily recognizable graph structures which can be computed in linear time. © 1994 by John Wiley & Sons, Inc.