Improved Algorithms for Detecting Negative Cost Cycles in Undirected Graphs

  • Xiaofeng Gu
  • Kamesh Madduri
  • K. Subramani
  • Hong-Jian Lai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5598)


In this paper, we explore the design of algorithms for the problem of checking whether an undirected graph contains a negative cost cycle (UNCCD). It is known that this problem is significantly harder than the corresponding problem in directed graphs. Current approaches for solving this problem involve reducing it to either the b-matching problem or the T-join problem. The latter approach is more efficient in that it runs in O(n 3) time on a graph with n vertices and m edges, while the former runs in O(n 6) time. This paper shows that instances of the UNCCD problem, in which edge weights are restricted to be in the range { − K··K} can be solved in O(n 2.75·logn) time. Our algorithm is basically a variation of the T-join approach, which exploits the existence of extremely efficient shortest path algorithms in graphs with integral positive weights. We also provide an implementation profile of the algorithms discussed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)zbMATHGoogle Scholar
  2. 2.
    Korte, B., Vygen, J.: Combinatorial Optimization. Algorithms and Combinatorics, vol. 21. Springer, New York (2000)zbMATHGoogle Scholar
  3. 3.
    Gabow, H.N.: Data structures for weighted matching and nearest common ancestors with linking. In: Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, pp. 434–443 (1990)Google Scholar
  4. 4.
    Shoshan, A., Zwick, U.: All pairs shortest paths in undirected graphs with integer weights. In: FOCS, pp. 605–615 (1999)Google Scholar
  5. 5.
    Gabow, H.N.: A scaling algorithm for weighted matching on general graphs. In: Proceedings 26th Annual Symposium of the Foundations of Computer Science, pp. 90–100. IEEE Computer Society Press, Los Alamitos (1985)Google Scholar
  6. 6.
    Demetrescu, C., Goldberg, A., Johnson, D.: 9th DIMACS implementation challenge – Shortest Paths (2005),
  7. 7.
    Gabow, H.: Implementation of algorithms for maximum matching on non-bipartite graphs. PhD thesis, Stanford University (1974)Google Scholar
  8. 8.
    Rothberg, E.: Implementation of H. Gabow’s weighted matching algorithm (1992),
  9. 9.
    Goldberg, A.V.: Scaling algorithms for the shortest paths problem. SIAM Journal on Computing 24, 494–504 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Goldberg, A.: Shortest path algorithms: Engineering aspects. In: ISAAC: 12th International Symposium on Algorithms and Computation, pp. 502–513 (2001)Google Scholar
  11. 11.
    Guo, L., Mukhopadhyay, S., Cukic, B.: Does your result checker really check? In: Dependable Systems and Networks, pp. 399–404 (2004)Google Scholar
  12. 12.
    Kratsch, D., McConnell, R.M., Mehlhorn, K., Spinrad, J.: Certifying algorithms for recognizing interval graphs and permutation graphs. In: SODA, pp. 158–167 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Xiaofeng Gu
    • 1
  • Kamesh Madduri
    • 2
  • K. Subramani
    • 3
  • Hong-Jian Lai
    • 1
  1. 1.Department of MathematicsWest Virginia UniversityMorgantownUSA
  2. 2.Computational Research DivisionLawrence Berkeley National LaboratoryBerkeleyUSA
  3. 3.LDCSEE, West Virginia UniversityMorgantownUSA

Personalised recommendations