Testing Acyclicity of Directed Graphs in Sublinear Time

  • Michael A. Bender
  • Dana Ron
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1853)

Abstract

This paper initiates the study of testing properties of directed graphs. In particular, the paper considers the most basic property of directed graphs - acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the adjacency matrix representation, most appropriate for dense graphs, a testing algorithm is developed that requires query and time complexity of O(l/∈2), where ∈ is a distance parameter independent of the size of the graph. The algorithm, which can probe the adjacency matrix of the graph, accepts every graph that is acyclic, and rejects, with probability at least 2/3, every graph whose adjacency matrix should be modified in at least e fraction of its entries so that it become acyclic. For the incidence list representation, most appropriate for sparse graphs, an Ω(V 1/3) lower bound is proved on the number of queries and the time required for testing, where V is the set of vertices in the graph. These results stand in contrast to what is known about testing acyclicity in undirected graphs.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Alon, E. Fischer, M. Krivelevich, and M Szegedy. Efficient testing of large graphs. In Proceedings of the Fortieth Annual Symposium on Foundations of Computer Science, pages 656–666, 1999.Google Scholar
  2. 2.
    N. Alon, M. Krivelevich, I. Newman, and M Szegedy. Regular languages are testable with a constant number of queries. In Proceedings of the Fortieth Annual Symposium on Foundations of Computer Science, pages 645–655, 1999.Google Scholar
  3. 3.
    S. Arora, A. Frieze, and H. Kaplan. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. In 37th Annual Symposium on Foundations of Computer Science, pages 21–30. IEEE, 14–16 October 1996.Google Scholar
  4. 4.
    S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and intractability of approximation problems. In Proceedings of the Thirty-Third Annual Symposium on Foundations of Computer Science, pages 14–23, 1992.Google Scholar
  5. 5.
    S. Arora and S. Safra. Probabilistic checkable proofs: A new characterization of NP. In Proceedings of the Thirty-Third Annual Symposium on Foundations of Computer Science, pages 1–13, 1992.Google Scholar
  6. 6.
    L. Babai, L. Fortnow, L. Levin, and M. Szegedy. Checking computations in polylogarithmic time. In Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, pages 21–31, 1991.Google Scholar
  7. 7.
    L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1(1):3–40, 1991.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Bender and D. Ron. Testing acyclicity of directed graphs in sublinear time. Full version of this paper. Available from http://www.eng.tau.ac.il/danar, 2000.
  9. 9.
    B. Berger and P. W. Shor. Tight bounds for the maximum acyclic subgraph problem. Journal of Algorithms, 25(1):1–18, October 1997.Google Scholar
  10. 10.
    M. Blum, M. Luby, and R. Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of the Association for Computing Machinery, 47:549–595, 1993.MATHMathSciNetGoogle Scholar
  11. 11.
    Y. Dodis, O. Goldreich, E. Lehman, S. Raskhodnikova, D. Ron, and A. Samorodnitsky. Improved testing algorithms for monotonocity. In Proceedings of RANDOM99, 1999.Google Scholar
  12. 12.
    F. Ergun, S. Kannan, S. R. Kumar, R. Rubinfeld, and M. Viswanathan. Spotcheckers. In Proceedings of the Thirty-Second Annual ACM Symposium on the Theory of Computing, pages 259–268, 1998.Google Scholar
  13. 13.
    G. Even, J. Naor, B. Schieber, and M. Sudan. Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica, 20, 1998.Google Scholar
  14. 14.
    U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy. Approximating clique is almost NP-complete. Journal of the Association for Computing Machinery, 43(2):268–292, 1996.MATHGoogle Scholar
  15. 15.
    A. Frieze and R. Kannan. Quick approximations of matrices. An extended abstract of some of this work appeared in FOCS96 under the title: The Regularity Lemma and approximation schemes for dense problems, 1997.Google Scholar
  16. 16.
    P. B. Gibbons and Y. Matias. New sampling-based summary statistics for improving approximate query answers. SIGMOD Record: Proc. ACM SIGMOD Int. Conf. Management of Data, 27(2):331–342, 2–4 June 1998.Google Scholar
  17. 17.
    P. B. Gibbons and Y. Matias. Synopsis data structures for massive data sets. DI-MACS: Series in Discrete Mathematics and Theoretical Computer Science: Special Issue on External Memory Algorithms and Visualization, A, 1999. to appear.Google Scholar
  18. 18.
    M. Goemans and D. Williamson. Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica, 18, 1998.Google Scholar
  19. 19.
    O. Goldreich, S. Goldwasser, E. Lehman, D. Ron, and A. Samordinsky. Testing monotonicity. To appear in Combinatorica. A preliminary (and weaker) version of this work appeared in FOCS98, 1999.Google Scholar
  20. 20.
    O. Goldreich, S. Goldwasser, and D. Ron. Property testing and its connection to learning and approximation. Journal of the Association for Computing Machinery, 45(4):653–750, 1998. An extended abstract appeared in FOCS96.MATHMathSciNetGoogle Scholar
  21. 21.
    O. Goldreich and D. Ron. Property testing in bounded degree graphs. In Proceedings of the Thirty-First Annual ACM Symposium on the Theory of Computing, pages 406–415, 1997.Google Scholar
  22. 22.
    O. Goldreich and D. Ron. A sublinear bipartite tester for bounded degree graphs. In Proceedings of the Thirty-Second Annual ACM Symposium on the Theory of Computing, 1998. To appear in Combinatorica.Google Scholar
  23. 23.
    R. Hassin and S. Rubinstein. Approximations for the maximum acyclic subgraph problem. Information Processing Letters, 51(3):133–140, August 1994.Google Scholar
  24. 24.
    D. S. Hochbaum and D. B. Shmoys. Using dual approximation algorithms for scheduling problems: Theoretical and practical results. Journal of the Association for Computing Machinery, 34(1):144–162, January 1987.Google Scholar
  25. 25.
    D. S. Hochbaum and D. B. Shmoys. A polynomial approximation scheme for machine scheduling on uniform processors: Using the dual approximation approach. SI AM Journal on Computing, 17(3):539–551, 1988.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    V. Kann. On the Approximability of NP-Complete Optimization Problems. PhD thesis, Department of Numberical Analysis and Computer Science, Royal Institute of Technology, Stockholm, 1992.Google Scholar
  27. 27.
    R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 85–103, New York, 1972. Plenum Press.Google Scholar
  28. 28.
    M. Kearns and D. Ron. Testing problems with sub-learning sample complexity. In Proceedings of the Eleventh Annual ACM Conference on Computational Learning Theory, pages 268–277, 1998.Google Scholar
  29. 29.
    C. Papadimitriou and M. Yannakakis. Optimization, approximization and complexity classes. Journal of Computer and System Sciences, 43:425–440, 1991.MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    M. Parnas and D. Ron. Testing the diameter of graphs. In Proceedings of Random99, pages 85–96, 1999.Google Scholar
  31. 31.
    R. Rubinfeld. Robust functional equations and their applications to program testing. In Proceedings of the Thirty-Fifth Annual Symposium on Foundations of Computer Science, 1994.Google Scholar
  32. 32.
    R. Rubinfeld and M. Sudan. Robust characterization of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252–271, 1996.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    P. D. Seymour. Packing directed circuits fractionally. Combinatorica, 15, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Michael A. Bender
    • 1
  • Dana Ron
    • 2
  1. 1.Department of Computer ScienceState University of New York at Stony BrookStony BrookUSA
  2. 2.Department of Electrical Engineering - SystemsTel Aviv UniversityRamat AvivIsrael

Personalised recommendations