Path problems in skew-symmetric graphs

Abstract

We study path problems in skew-symmetric graphs. These problems generalize the standard graph reachability and shortest path problems. We establish combinatorial solvability criteria and duality relations for the skew-symmetric path problems and use them to design efficient algorithms for these problems. The algorithms presented are competitive with the fastest algorithms for the standard problems.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    R. K. Ahuja, K. Mehlhorn, J. B. Orlin, andR. E. Tarjan: Faster Algorithms for the Shortest Path Problem,Technical Report CS-TR-154-88, Department of Computer Science, Princeton University, 1988.

  2. [2]

    R. E. Bellman: On a Routing Problem,Quart. Appl. Math. 16 (1958), 87–90.

    Google Scholar 

  3. [3]

    C. Berge: Two Theorems in Graph Theory,Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 842–844.

    Google Scholar 

  4. [4]

    T. H. Cormen, C. E. Leiserson, andR. L. Rivest:Introduction to Algorithms, (MIT Press, Cambridge, MA, 1990).

    Google Scholar 

  5. [5]

    E. W. Dijkstra: A Note on Two Problems in Connection with Graphs,Numer. Math. 1 (1959), 269–271.

    Google Scholar 

  6. [6]

    J. Edmonds: Maximum Matchings and a Polyhedron with 0,1-vertices,J. Res. Nat. Bur. Stand. 96B (1965), 125–130.

    Google Scholar 

  7. [7]

    J. Edmonds: Paths, Trees and Flowers,Canada J. Math. 17 (1965), 449–467.

    Google Scholar 

  8. [8]

    J. Edmonds, andE. L. Johnson: Matching, a Well-Solved Class of Integer Linear Programs, In: R. Guy, H. Haneni, and J. Schönhein, editors,Combinatorial Structures and Their Applications (Gordon and Breach, NY, 1970), 89–92.

    Google Scholar 

  9. [9]

    L. R. Ford, Jr., andD. R. Fulkerson:Flows in Networks (Princeton Univ. Press, Princeton, NJ, 1962).

    Google Scholar 

  10. [10]

    M. L. Fredman, andR. E. Tarjan: Fibonacci Heaps and Their Uses in Improved Network Optimization Algorithms,J. Assoc. Comput. Mach. 34 (1987), 596–615.

    Google Scholar 

  11. [11]

    H. N. Gabow, S. N. Maheshwari, andL. Osterweil: On Two Problems in the Generation of Program Test Paths,IEEE Trans. on Software Eng. SE-2 (1976), 227–231.

    Google Scholar 

  12. [12]

    H. N. Gabow, andR. E. Tarjan: A Linear-Time Algorithm for a Special Case of Disjoint Set Union,J. Comp. and Syst. Sci. 30 (1985), 209–221.

    Google Scholar 

  13. [13]

    A. V. Goldberg: Scaling Algorithms for the Shortest Paths Problem, In:Proc. 4th ACM-SIAM Symposium on Discrete Algorithms, 1993, 222–231.

  14. [14]

    A. V. Goldberg, andA. V. Karzanov: Maximum Flows in Skew-Symmetric Graphs,Technical Report (Stanford University, Stanford, 1995).

    Google Scholar 

  15. [15]

    A. V. Goldberg, andA. V. Karzanov: Minimum Cost Flows in Skew-Symmetric Graphs, in preparation.

  16. [16]

    M. Grötschel, L. Lovász, andA. Schrijver:Geometric Algorithms and Combinatorial Optimization (Springer Verlag, 1988).

  17. [17]

    A. V. Karzanov: An algorithm for determining a maximum packing of oddterminus cuts and its applications, in:Studies in Applied Graph Theory (A. S. Alekseev, ed., Nauka, Novosibirsk, 1986), 126–140; in Russian; English translation inAmer. Math. Soc. Translations, Ser. 2,158 (1994), 57–70.

    Google Scholar 

  18. [18]

    A. V. Karzanov: A Minimum Cost Maximum Multiflow Problem, In:Combinatorial Methods for Flow Problems (Inst. for Systems Studies, Moscow, volume 4, 1979), 138–156; In Russian.

    Google Scholar 

  19. [19]

    A. V. Karzanov: Minimum Cost Multiflows in Undirected Networks,Mathematical Programming 66 (3) (1994), 313–325.

    Google Scholar 

  20. [20]

    L. Lovász: The Factorization of Graphs II,Acta Math. Hung. 23 (1972), 223–246.

    Google Scholar 

  21. [21]

    L. Lovász, andM. D. Plummer:Matching Theory (Akadémiai Kiadó, Budapest, 1986).

    Google Scholar 

  22. [22]

    E. F. Moore: The Shortest Path Through a Maze, In:Proc. of the Int. Symp. on the Theory of Switching (Harvard University Press, 1959), 285–292.

  23. [23]

    R. E. Tarjan: Efficiency of a Good but not Linear Set Union Algorithm,J. Assoc. Comput. Mach. 22 (1975), 1975.

    Google Scholar 

  24. [24]

    W. T. Tutte: The Factorization of Linear Graphs,J. London Math. Soc. 22 (1947), 107–111.

    Google Scholar 

  25. [25]

    W. T. Tutte: Antisymmetrical Digraphs,Canada J. Math. 19 (1967), 1101–1117.

    Google Scholar 

  26. [26]

    H. Yinnone: On Paths Avoiding Forbidden Pairs of Vertices in a Graph, submitted toDiscrete Appl. Math.

Download references

Author information

Affiliations

Authors

Additional information

This research was done while the first author was at Stanford University Computer Science Department, supported in part by ONR Office of Naval Research Young Investigator Award N00014-91-J-1855, NSF Presidential Young Investigator Grant CCR-8858097 with matching funds from AT&T, DEC, and 3M, and a grant from Powell Foundation.

This research was done while the second author was visiting Stanford University Computer Science Department and supported by the above mentioned NSF and Powell Foundation Grants.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Goldberg, A.V., Karzanov, A.V. Path problems in skew-symmetric graphs. Combinatorica 16, 353–382 (1996). https://doi.org/10.1007/BF01261321

Download citation

Mathematics Subject Classification (1991)

  • 05 C