Flow symmetry and algebraic flows

Abstract

Analgebraic flow for a digraphD=(V, A) is a generalization of acirculation forD in which the operation of addition is replaced by a binary operation defined over a commutative semigroup. A substantial literature exists in which flow-theory has been studied in this more general setting. For example, Hamacher has generalized the classical max-flow min-cut theorem to algebraic flows. In this paper, we show thatx is an algebraic flow if and only if for each pair of distinct verticess andt, the value of a maximum (s, t) algebraic flow with capacitiesx is equal to the value of a maximum (t, s) algebraic flow with capacitiesx. This characterization, which we callflow-symmetry, is a common generalization of two previous flow-symmetric results that have appeared in the literature. First, Lovász, by proving a conjecture of Kotzig, showed that flow-symmetry holds for the usual semigroup operation of addition of non-negative reals. That is, a vectorx≥0 defined on the arc setA is a circulation forD if and only if for each pair of distinct verticess andt the value of a maximum (s, t) flow inD with capacitiesx equals the value of a maximum (t, s) flow inD with capacitiesx. Second, in a previous paper, we showed that the analogous result holds for the semigroup in which the summation operator is replaced by the maximization operator. That is,x is amax-balanced flow if and only if for each pair of distinct verticess andt, the value of a maximum bottleneck (s, t) path inD with capacitiesx equals the value of a maximum bottleneck (t, s) path inD with capacitiesx. In this paper, we show that these results are each special cases of our characterization of an algebraic flow.