Skip to main content
SpringerLink
Log in

Formalizing the Edmonds-Karp Algorithm

  • Conference paper
  • First Online:
Interactive Theorem Proving (ITP 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9807))

Included in the following conference series:

Abstract

We present a formalization of the Ford-Fulkerson method for computing the maximum flow in a network. Our formal proof closely follows a standard textbook proof, and is accessible even without being an expert in Isabelle/HOL — the interactive theorem prover used for the formalization. We then use stepwise refinement to obtain the Edmonds-Karp algorithm, and formally prove a bound on its complexity. Further refinement yields a verified implementation, whose execution time compares well to an unverified reference implementation in Java.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Section 8.1 provides a detailed discussion.

  2. 2.

    With \(u=v\), this also implies that there are no self loops.

  3. 3.

    Up to this point, the formalization models capacities as linearly ordered integral domains, which subsume reals, rationals, and integers. Thus, we could chose any executable number representation here.

References

  1. Back, R.-J.: On the correctness of refinement steps in program development. Ph.D. thesis, Department of Computer Science, University of Helsinki (1978)

    Google Scholar 

  2. Back, R.-J., von Wright, J.: Refinement Calculus - A Systematic Introduction. Springer, New York (1998)

    Book  MATH  Google Scholar 

  3. Ballarin, C.: Interpretation of locales in Isabelle: theories and proof contexts. In: Borwein, J.M., Farmer, W.M. (eds.) MKM 2006. LNCS (LNAI), vol. 4108, pp. 31–43. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  4. Bertot, Y., Castran, P., Proving, I.T., Development, P.: Coq’Art The Calculus of Inductive Constructions, 1st edn. Springer (2010)

    Google Scholar 

  5. Bulwahn, L., Krauss, A., Haftmann, F., Erkök, L., Matthews, J.: Imperative functional programmingwith Isabelle/HOL. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 134–149. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  6. Charguéraud, A.: Characteristic formulae for the verification of imperative programs. In: ICFP, pp. 418–430. ACM (2011)

    Google Scholar 

  7. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press (2009)

    Google Scholar 

  8. Dinitz, Y.: Dinitz’ algorithm: the original version and Even’s version. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Theoretical Computer Science. LNCS, vol. 3895, pp. 218–240. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  9. Edmonds, J., Karp, R.M.: Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19(2), 248–264 (1972)

    Article  MATH  Google Scholar 

  10. Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Can. J. Math. 8(3), 399–404 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  11. Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Greenaway, D.: Automated proof-producing abstraction of C code. Ph.D. thesis, CSE, UNSW, Sydney, Australia (2015)

    Google Scholar 

  13. Greenaway, D., Andronick, J., Klein, G.: Bridging the gap: automatic verified abstraction of C. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 99–115. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  14. Haftmann, F.: Code generation from specifications in higher order logic. Ph.D. thesis, Technische Universität München (2009)

    Google Scholar 

  15. Haftmann, F., Nipkow, T.: Code generation via higher-order rewrite systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) FLOPS 2010. LNCS, vol. 6009, pp. 103–117. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  16. Krauss, A.: Recursive definitions of monadic functions. In: Proceedings of PAR, vol. 43, pp. 1–13 (2010)

    Google Scholar 

  17. Lammich, P.: Refinement for monadic programs. In: Archive of Formal Proofs, Formal proof development (2012). http://afp.sf.net/entries/Refine_Monadic.shtml

  18. Lammich, P.: Verified efficient implementation of Gabow’s strongly connected component algorithm. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 325–340. Springer, Heidelberg (2014)

    Google Scholar 

  19. Lammich, P.: Refinement to Imperative/HOL. In: Urban, C., Zhang, X. (eds.) ITP 2015. LNCS, vol. 9236, pp. 253–269. Springer, Heidelberg (2015)

    Google Scholar 

  20. Lammich, P.: Refinement based verification of imperative data structures. In: CPP, pp. 27–36. ACM (2016)

    Google Scholar 

  21. Lammich, P., Meis, R.: A separation logic framework for Imperative HOL. Archive of Formal Proofs, Formal proof development, Nov. 2012. http://afp.sf.net/entries/Separation_Logic_Imperative_HOL.shtml

  22. Lammich, P., Tuerk, T.: Applying data refinement for monadic programs to Hopcroft’s algorithm. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 166–182. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  23. Lee, G.: Correctnesss of Ford-Fulkersons maximum flow algorithm. Formalized Math. 13(2), 305–314 (2005)

    Google Scholar 

  24. Lee, G., Rudnicki, P.: Alternative aggregates in Mizar. In: Kauers, M., Kerber, M., Miner, R., Windsteiger, W. (eds.) MKM/CALCULEMUS 2007. LNCS (LNAI), vol. 4573, pp. 327–341. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  25. Matuszewski, R., Rudnicki, P.: Mizar: the first 30 years. Mechanized Math. Appl. 4(1), 3–24 (2005)

    Google Scholar 

  26. MLton Standard ML compiler. http://mlton.org/

  27. Nipkow, T., Paulson, L.C., Wenzel, M. (eds.): Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  28. Nordhoff, B., Lammich, P.: Formalization of Dijkstra’s algorithm. Archive of Formal Proofs, Formal proof development, Jan. 2012. http://afp.sf.net/entries/Dijkstra_Shortest_Path.shtml

  29. Noschinski, L.: Formalizing graph theory and planarity certificates. Ph.D. thesis, Fakultät für Informatik, Technische Universität München, November 2015

    Google Scholar 

  30. Reynolds, J.C.: Separation logic: A logic for shared mutable data structures. In: Proceedings of Logic in Computer Science (LICS), pp. 55–74. IEEE (2002)

    Google Scholar 

  31. Sedgewick, R., Wayne, K.: Algorithms, 4th edn. Addison-Wesley (2011)

    Google Scholar 

  32. Wenzel, M.: Isar - A generic interpretative approach to readable formal proof documents. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 167–184. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  33. Wirth, N.: Program development by stepwise refinement. Commun. ACM 14(4), 221–227 (1971)

    Article  MATH  Google Scholar 

  34. Zwick, U.: The smallest networks on which the Ford-Fulkerson maximum flow procedure may fail to terminate. Theor. Comput. Sci. 148(1), 165–170 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technische Universität München, Munich, Germany

    Peter Lammich & S. Reza Sefidgar

Authors
  1. Peter Lammich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Reza Sefidgar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Peter Lammich or S. Reza Sefidgar .

Editor information

Editors and Affiliations

  1. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Jasmin Christian Blanchette

  2. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Stephan Merz

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Lammich, P., Sefidgar, S.R. (2016). Formalizing the Edmonds-Karp Algorithm. In: Blanchette, J., Merz, S. (eds) Interactive Theorem Proving. ITP 2016. Lecture Notes in Computer Science(), vol 9807. Springer, Cham. https://doi.org/10.1007/978-3-319-43144-4_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-43144-4_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-43143-7

  • Online ISBN: 978-3-319-43144-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation