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.
Similar content being viewed by others
References
Burkard RE, Cuninghame-Green RA, Zimmermann U (1984) Algebraic and combinatorial methods in operations research. Annals of Discrete Mathematics 19. North Holland. Amsterdam
Burkard RE, Zimmermann U (1982) Combinatorial optimization in linearly ordered semimodules: A survey. Modern Applied Mathematics—Optimization and Operations Research. North Holland. Amsterdam
Edmonds J, Fulkerson DR (1970) Bottleneck extrema. Journal of Combinatorial Theory 8:299–306
Hamacher H (1980) Maximal algebraic flows: algorithms and examples. Pape U. ed. Discrete Structures and Algorithms. Hanser Verlag. Munich: 153–166
Hartmann M, Schneider MH (1992) An analogue of Hoffman's circulation conditions for max-balanced flows. Mathematical Programming, 57:459–476
Hartmann M, Schneider MH (1991) Max-balanced flows in oriented matroids. To appear in Discrete Mathematics
Lovász L (1973) Connectivity in digraphs. Journal of Combinatorial Theory B 15:174–177
Schneider H, Schneider MH (1991) Max-balancing weighted directed graphs and matrix scaling. Mathematics of Operations Research 16:208–220
Zimmermann U (1981) Linear and combinatorial optimization in ordered algebraic structures. Annals of Discrete Mathematics 10. North Holland. Amsterdam
Author information
Authors and Affiliations
Additional information
Research supported in part by NSF graph MS-89-05645.
Research was, in part, conducted at Johns Hopkins University and supported by NSF grant ECS 87-18971.
Rights and permissions
About this article
Cite this article
Hartmann, M., Schneider, M.H. Flow symmetry and algebraic flows. ZOR - Methods and Models of Operations Research 38, 261–267 (1993). https://doi.org/10.1007/BF01416607
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01416607