Journal of Combinatorial Optimization

, Volume 19, Issue 3, pp 369–393 | Cite as

A fast exact algorithm for the problem of optimum cooperation and the structure of its solutions

  • Diana FanghänelEmail author
  • Frauke Liers


Given a graph G=(V,E) with edge weights w e ∈ℝ, the optimum cooperation problem consists in determining a partition of the graph that maximizes the sum of weights of the edges with nodes in the same class plus the number of the classes of the partition. The problem is also known in the literature as the optimum attack problem in networks. Furthermore, a relevant physics application exists.

In this work, we present a fast exact algorithm for the optimum cooperation problem. Algorithms known in the literature require |V|−1 minimum cut computations in a corresponding network. By theoretical considerations and appropriately designed heuristics, we considerably reduce the numbers of minimum cut computations that are necessary in practice. We show the effectiveness of our method by presenting results on instances coming from the physics application. Furthermore, we analyze the structure of the optimal solutions.


Optimum attack problem Submodular function minimization Potts glass with many states 


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  1. Anglès d’Auriac JC, Iglói F, Preissmann M, Sebő A (2002) Optimal cooperation and submodularity for computing Potts’ partition functions with a large number of states. J Phys A, Math Gen 35:6973–6983 zbMATHCrossRefGoogle Scholar
  2. Baïou M, Barahona F, Mahjoub R (2000) Separation of partition inequalities. Math Operat Res 25(2):243–254 zbMATHCrossRefGoogle Scholar
  3. Barahona F (1992) Separating from the dominant of the spanning tree polytope. Oper Res Lett 12:201–203 zbMATHCrossRefMathSciNetGoogle Scholar
  4. Cunningham WH (1985) Optimal attack and reinforcement of a network. J Assoc Comput Mach 32(3):549–561 zbMATHMathSciNetGoogle Scholar
  5. Dempe S, Schreier H (2006) Operations Research—Deterministische Modelle und Methoden. Teubner, Wiesbaden zbMATHGoogle Scholar
  6. Fujishige S (2005) Submodular functions and optimization. Annals of discrete mathematics, vol 58. Elsevier, Amsterdam zbMATHGoogle Scholar
  7. Goldberg AV, Tarjan RE (1988) A new approach to the maximum-flow problem. J ACM 35(4):921–940 zbMATHCrossRefMathSciNetGoogle Scholar
  8. Grätzer G (1978) General lattice theory. Birkhäuser, Basel Google Scholar
  9. Grötschel M, Lovász L, Schrijver A (1981) The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2):169–197 zbMATHCrossRefMathSciNetGoogle Scholar
  10. Grötschel M, Lovász L, Schrijver A (1988) Geometric algorithms and combinatorial optimization. Springer, Berlin zbMATHGoogle Scholar
  11. Hartmann AK, Rieger H (2002) Optimization algorithms in physics. Wiley-VHC, Berlin zbMATHGoogle Scholar
  12. Hartmann AK, Rieger H (2004) New optimization algorithms in physics. Wiley-VHC, Berlin zbMATHCrossRefGoogle Scholar
  13. Iwata S, Fleischer L, Fujishige S (2001) A combinatorial strongly polynomial algorithm for minimizing submodular functions. J ACM 48(4):761–777 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Juhasz R, Rieger H, Iglói F (2001) The random-bond Potts model in the large-q limit. Phys Rev E 64:056122 CrossRefGoogle Scholar
  15. Jünger M, Rinaldi G, Thienel S (2000) Practical performance of efficient minimum cut algorithms. Algorithmica 26:172–195 zbMATHCrossRefMathSciNetGoogle Scholar
  16. McCormick ST (2005) Submodular function minimization. In: Aardal K et al. (eds) Discrete optimization. Handbooks in operations research and management science, vol 12. Elsevier, Amsterdam, pp 321–391 CrossRefGoogle Scholar
  17. Open Graph Drawing Framework (2007).
  18. Preissmann M, Sebő A (2009) Graphic submodular function minimization: a graphic approach and applications. In: Cook W, Lovász L, Vygen J (eds) Research trends in combinatorial optimization. Springer, Berlin, pp 365–386 CrossRefGoogle Scholar
  19. Schrijver A (2000) A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J Comb Theory B 80:346–355 zbMATHCrossRefMathSciNetGoogle Scholar
  20. Topkis DM (1998) Supermodularity and complementarity. Princeton University Press, Princeton Google Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institut für InformatikUniversität zu KölnKölnGermany

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