Combinatorial Optimization: The Interplay of Graph Theory, Linear and Integer Programming Illustrated on Network Flow

  • Annegret K. WaglerEmail author
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


Combinatorial optimization is one of the fields in mathematics with an impressive development in recent years, driven by demands from applications where discrete models play a role. Here, we intend to give a comprehensive overview of basic methods and paradigms, in particular the beautiful interplay of methods from graph theory, geometry, and linear and integer programming related to combinatorial optimization problems. To understand the underlying framework and the interrelationships more clearly, we illustrate the theoretical results and methods with the help of flows in networks as running example. This includes, on the one hand, a combinatorial algorithm for finding a maximum flow in a network, combinatorial duality and the max-flow min-cut theorem as one of the fundamental combinatorial min–max relations. On the other hand, we discuss solving the network flow problem as a linear program with the help of the simplex method, linear programming duality and the dual program for network flow. Finally, we address the problem of integer network flows, ideal formulations for integer linear programs and consequences for the network flow problem.


Network flow Max-flow min-cut theorem Linear programming Duality Integer linear programming Unimodularity 


  1. 1.
    Balas, E., Saxena, A.: Optimizing over the split closure. Math. Program. 113, 219–240 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Basu, A., Cornuéjols, G., Margot, M.: Intersection cuts with infinite split rank. Math. Oper. Res. 37, 21–40 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertsimas, D., Weismantel, R.: Optimization over Integers. Dynamic Ideas, Belmont (2005) Google Scholar
  4. 4.
    Bland, R.G.: New finite pivoting rules for the simplex method. Math. Oper. Res. 2, 103–107 (1977) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math. 4, 305–337 (1973) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Corner polyhedron and intersection cuts. Surv. Oper. Res. Manag. Sci. 16, 105–120 (2011) Google Scholar
  7. 7.
    Dantzig, G.B.: Maximization of a linear function of variables subject to linear inequalities. In: Koopmans, T.C. (ed.) Activity Analysis of Production and Allocation, pp. 339–347. Wiley, New York (1951) Google Scholar
  8. 8.
    Dantzig, G.B.: Notes on Linear Programming. RAND Corporation (1953) Google Scholar
  9. 9.
    Dantzig, G.B.: The diet problem. Interfaces 20, 43–47 (1990). The Practice of Mathematical Programming CrossRefGoogle Scholar
  10. 10.
    Dantzig, G.B., Thapa, M.N.: Linear Programming 2: Theory and Extensions. Springer, Berlin (2003) Google Scholar
  11. 11.
    Del Pia, A., Wagner, C., Weismantel, R.: A probabilistic comparison of the strength of split, triangle, and quadrilateral cuts. Oper. Res. Lett. 39, 234–240 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Edmonds, J., Giles, R.: A min–max relation for submodular functions on graphs. In: Hammer, P.L., Johnson, E.L., Korte, B.H., Nemhauser, G.L. (eds.) Studies in Integer Programming, Proceedings of the Workshop on Integer Programming, Bonn, 1975, pp. 185–204 (1977) CrossRefGoogle Scholar
  13. 13.
    Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19, 248–264 (1972) CrossRefzbMATHGoogle Scholar
  14. 14.
    Farkas, G.: A Fourier-féle mechanikai elv alkamazásai. Math. Termśzettudományi Értesítö 12, 457–472 (1894) Google Scholar
  15. 15.
    Farkas, G.: Über die Theorie der Einfachen Ungleichungen. J. Reine Angew. Math. 124, 1–27 (1902) Google Scholar
  16. 16.
    Ford, L.R., Fulkerson, D.R.: Maximum flow through a network. Can. J. Math. 8, 399–404 (1956) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ford, L.R., Fulkerson, D.R.: Network Flow Theory. Princeton Press, Princeton (1962) Google Scholar
  18. 18.
    Gomory, R.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64, 275–278 (1958) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988) CrossRefzbMATHGoogle Scholar
  20. 20.
    Hoffman, A.J., Kruskal, J.B.: Integral boundary points of convex polyhedra. In: Kuhn, H.W., Tucker, A.W. (eds.) Linear: Inequalities and Related Systems. Annals of Mathematics Studies, vol. 38, pp. 223–246. Princeton University Press, Princeton (1956) Google Scholar
  21. 21.
    Kannan, R., Monma, C.L.: On the computational complexity of integer programming problems. Lect. Notes Econ. Math. Syst. 157, 161–172 (1978) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Klee, V., Minty, G.J.: How good is the simplex algorithm? In: Shisha, O. (ed.) Inequalities III, pp. 159–175. Academic Press, New York (1972) Google Scholar
  23. 23.
    Minkowski, H.: Allgemeine Lehrsätze über konvexe Polyeder. Ges. Wiss. Göttingen 198–219 (1897) Google Scholar
  24. 24.
    Nemhauser, G.L., Wolsey, L.A.: Integer Programming and Combinatorial Optimization. Wiley-Interscience, New York (1998) Google Scholar
  25. 25.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986) zbMATHGoogle Scholar
  26. 26.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003) Google Scholar
  27. 27.
    Weyl, H.: Elementare Theorie der konvexen Polyeder. Comment. Math. Helv. 7, 290–306 (1935) MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Laboratoire d’Informatique, de Modélisation et d’Optimisation des Systèmes (LIMOS)/CNRSUniversité Blaise Pascal (Clermont-Ferrand II)Aubière CedexFrance

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