Substitutes, Complements and Ripples in Network Flows

Abstract

We study the qualitative variation of minimum-cost network-flows and their associated costs with arc parameters. Demands are given at each node, flow is conserved, each arc's parameter lies in a lattice, and the flow cost is real or infinity-valued. Our main results are roughly as follows. If the flow cost is arc-additive, the problem can be decomposed into independent problems on each biconnected component of the graph. If also the flow cost is convex in the flow, the magnitude of the change in the optimal flow in arc a resulting from changing arc b's parameter diminishes the less biconnected a is to b. Arc a is less-biconnected-to b than is arc d if every simple cycle containing a and b also contains d. This relation is a quasi order with two distinct arcs being equivalent if and only if deleting them disconnects the graph. Hie Hasse diagram of the partial order of the induced equivalence classes is a tree with all arcs directed towards the class containing b. Two arcs are complements (resp., substitutes) if every simple cycle containing both orients them in the same (resp., opposite) way. For example, two arcs are conformal, i.e., complements or substitutes, if they are either incident, or lie on a common face of a planar graph, or (as Dirac [Dirac, G. A. 1952. A property of 4-chromatic graphs and some remarks on critical graphs. J. London Math. Soc. 27 85–92.] and Duffin [Duffin, R. J. 1965. Topology of series-parallel networks. J. Math. Anal. Appl. 10 303–318.] have shown) lie in a series-parallel graph. If also each arc cost is (lattice) subadditive, then the optimal flow in arc a is nondecreasing (resp., nonincreasing) in the parameter of each complement (resp., substitute) of a. If moreover each parameter lies in a chain, then the minimum cost is superadditive in the parameters of a set of substitutes. Suppose in addition that the arc parameters are real and the arc costs are doubly subadditive. Then the absolute difference of the optimal flows in an arc corresponding to two monotonically-step-connected parameter vectors does not exceed the sum of the absolute differences of their elements. If further the arc costs are affine between successive integers and the difference between the two parameter vectors is a unit vector, then the difference between corresponding integer optimal flows is a unit simple circulation. This fact leads to a parametric algorithm for finding optimal flows. Finally, suppose instead that the flow cost is a sum of a subadditive flow cost for arcs in a set S of complements and a flow cost for arcs not in S that is arc-additive and convex in the flows therein. Then the optimal flow in each arc in S is nondecreasing in the parameters of those arcs, and the minimum cost is subadditive therein. Moreover, the optimal flow in each arc a not in S is nondecreasing (resp., nonincreasing) in the parameters of arcs in S if a is a complement (resp., substitute) of every arc in S.