Abstract
We give a formulation of the maximum flow problem as an integer programming problem in the standard form. We characterize elementary vectors of the kernel lattice of the matrix coefficient in our formulation in terms of the combinatorial property of the graph, such as circuits and s–t paths, and we determine the universal Gröbner basis of the toric ideal associated with the maximum flow problem. Next, we examine the kernel lattice of the reduced incidence matrix of the digraph. Under suitable assumptions for the digraph, we prove that it is generated by all incidence vectors of s–t directed paths, which yield the generator of the toric ideal associated with the reduced incidence matrix of the digraph.
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References
Ahuja K., Magnanti L., Orlin B.: Network Flows. Prentice-Hall, New Jersey (1993)
Bachem A., Kern W.: Linear Programming Duality. Springer, Berlin (1992)
Bayer D., Popescu S., Sturmfels B.: Syzygies of unimodular Lawrence ideals. Journal fuer die Reine und Angewandte Mathematik 5344, 169–186 (2001)
Cox D., Little J., O’shea D.: Ideals, Varieties, and Algorithms, 2nd edn. Springer, New York (1996)
Cox D., Little J., O’shea D.: Using Algebraic Geometry. Springer, New York (1998)
Hosten, S., Sturmfels, B.: An implementation of Gröbner bases for integer programming. Lecture Notes in Computer Science, vol. 920, pp. 267–276. Springer (1995)
Jungnickel D.: Graphs, Networks and Algorithm, 2nd edn. Springer, Berlin (2005)
Korte B., Vygen J.: Combinatorial Optimization, 2nd edn. Springer, Berlin (2001)
Miyagi, N.: Toric ideal associated with the maximum flow problem. Master’s thesis (2008) (in Japanese)
Sturmfels B.: Gröbner Bases and Convex Polytopes. American Mathematical Society, Providence (1996)
Sturmfels B., Weismantel R., Ziegler G.: Gröbner bases of lattices, corner polyhedra, and integer programming. Beiträge zur Algebra und Geometrie 36, 281–298 (1995)
Weismantel R.: Test sets of integer programs. Math. Methods Oper. Res. 47, 1–37 (1998)
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Watanabe, S., Watanabe, Y. & Ikegami, D. Universal Gröbner basis associated with the maximum flow problem. Japan J. Indust. Appl. Math. 30, 39–50 (2013). https://doi.org/10.1007/s13160-012-0080-2
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DOI: https://doi.org/10.1007/s13160-012-0080-2