Skip to main content
SpringerLink
Log in

Local elimination algorithms for solving sparse discrete problems

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The class of local elimination algorithms is considered that make it possible to obtain global information about solutions of a problem using local information. The general structure of local elimination algorithms is described that use neighborhoods of elements and the structural graph describing the problem structure; an elimination algorithm is also described. This class of algorithms includes local decomposition algorithms for discrete optimization problems, nonserial dynamic programming algorithms, bucket elimination algorithms, and tree decomposition algorithms. It is shown that local elimination algorithms can be used for solving optimization problems.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Yu. I. Zhuravlev, Selected Works (Magistr, Moscow, 1998) [in Russian].

    Google Scholar 

  2. Yu. I. Zhuravlev and G. F. Losev, “Neighborhoods in Discrete Mathematics Problems,” Kibern. Sistemn. Anal., No. 2, 32–41 (1995).

  3. I. V. Sergienko and V. P. Shilo, Discrete Optimization: Problems, Methods, Studies (Naukova Dumka, Kiev, 2003) [in Russian].

    Google Scholar 

  4. R. Dechter, “Bucket Elimination: A Unifying Framework for Reasoning,” Artif. Intell. 113, 41–85 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  5. R. E. Neapolitan, Probabilistic Reasoning in Expert Systems (Wiley, New York, 1990).

    Google Scholar 

  6. J. Pearl, Probabilistic Reasoning in Intelligent Systems (Morgan Kaufmann, San Mateo, 1998).

    Google Scholar 

  7. C. Beeri, R. Fagin, D. Maier, and M. Yannakakis, “On the Desirability of Acyclic Database Schemes,” J. ACM 30, 479–513 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  8. D. J. Rose, “A Graph-Theoretic Study of the Numerical Solution of Sparse Positive Definite Systems of Linear Equations,” in Graph Theory and Computing, Ed. by R. C. Read (Academic, New York, 1972), pp. 183–217.

    Google Scholar 

  9. Supply Chain Optimization: Product/Process Design, Facilities Location and Flow Control, Ser. Applied Optimization, XVI (Springer, New York, 2005), Vol. 94.

  10. S. A. Cook, “The Complexity of Theorem-Proving Procedures,” in Proc. 3rd Ann. ACM Symp. on Theory Comput. Mach. (New York, 1971), pp. 151–158.

  11. S. L. Lauritzen and D. J. Spiegelhalter, “Local Computations with Probabilities on Graphical Structures and Their Application to Expert Systems,” J. Roy. Statist. Soc. 50, 205–247 (1988).

    MathSciNet  Google Scholar 

  12. M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic, New York, 1980).

    MATH  Google Scholar 

  13. B. Courcelle, “Graph Rewriting: An Algebraic and Logic Approach,” in Handbook of Theoretical Computer Science, vol. B, Ed. by J. Van Leeuwen (Elsevier, Amsterdam, 1990), pp. 193–242.

    Google Scholar 

  14. S. Arnborg, D. G. Corneil, and A. Proskurowski, “Complexity of Finding Embeddings in a k-Tree,” SIAM J. Alg. Disc. Meth. 8, 277–284 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  15. O. A. Shcherbina, “On Local Algorithms for Solving Quasi-Block (Staircase) Discrete Programming Problems,” in Problemy kibernetiki (Nauka, Moscow, 1983), No. 40, pp. 171–200 [in Russian].

    Google Scholar 

  16. O. A. Shcherbina, “Local Algorithms for Block Treelike Discrete Programming Problems,” Zh. Vychisl. Mat. Mat. Fiz. 25, 1143–1154 (1985).

    MathSciNet  MATH  Google Scholar 

  17. O. A. Shcherbina, “On A Nonserial Modification of a Local Decomposition Algorithm for Discrete Optimization Problems,” Dinamich. Sistemy, No. 19, 179–190 (2005).

  18. O. A. Shcherbina, “Elimination Decomposition Algorithms in Discrete Optimization Problems,” Tavrich. Vestn. Informatiki Mat., No. 2, 28–41 (2006).

  19. O. A. Shcherbina, “Local Algorithms and Tree Decomposition in Discrete Optimization,” Dinamich. Sistemy, No. 20, 89–103 (2006).

  20. U. Bertele and F. Brioschi, Nonserial Dynamic Programming (Academic, New York, 1972).

    MATH  Google Scholar 

  21. F. Harary, R. Z. Norman, and D. Cartwright, Structural Models: An Introduction to the Theory of Directed Graphs (Wiley, New York, 1965).

    MATH  Google Scholar 

  22. P. R. Amestoy, T. A. Davis, and I. S. Duff, “An Approximate Minimum Degree Ordering Algorithm,” SIAM J. Matrix Analys. Appl. 17, 886–905 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  23. V. A. Emelichev, O. I. Mel’nikov, V. I. Sarvanov, and R. I. Tyshkevich, Lectures on Graph Theory (Nauka, Moscow, 1990) [in Russian].

    MATH  Google Scholar 

  24. P. Heggernes, “Minimal Triangulations of Graphs: A Survey,” Discrete Math. 305, 297–317 (2006).

    Article  MathSciNet  Google Scholar 

  25. V. A. Evstigneev and V. N. Kas’yanov, Dictionary of Graph Theory in Information and Computer Science (Nauka, Novosibirsk, 1999) [in Russian].

    Google Scholar 

  26. S. Parter, “The Use of Linear Graphs in Gauss Elimination,” SIAM Rev. 3, 119–130 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  27. D. R. Fulkerson and O. A. Gross, “Incidence Matrices and Interval Graphs,” Pacific J. Math. 15, 835–855 (1965).

    MathSciNet  MATH  Google Scholar 

  28. M. Yannakakis, “Computing the Minimum Fill-in Is NP-Complete,” SIAM J. Matrix Analys. Appl. 2, 77–79 (1981).

    MathSciNet  MATH  Google Scholar 

  29. N. Robertson and P. Seymour, “Graph Minors II. Algorithmic Aspects of Treewidth,” J. Algorithms 7, 309–322 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  30. H. L. Bodlaender, “Discovering Treewidth,” in Proc. SOFSEM, Lect. Notes Comput. Sci. 3381 (Springer, Berlin, 2006), pp. 1–16.

    Google Scholar 

  31. R. Dechter and Y. El Fattah, “Topological Parameters for Time-Space Tradeoff,” Artif. Intell. 125(1–2), 93–118 (2001).

    Article  MATH  Google Scholar 

  32. P. Heggernes, “Treewidth, Partial k-Trees, and Chordal Graphs,” http://www.ii.uib.no/:_pinar/chordal.pdf.

  33. G. A. Dirac, “On Rigid Circuit Graphs,” Ann. Math. Sem. (Hamburg Univ., Hamburg, 1961), Vol. 25, pp. 71–76.

    Google Scholar 

  34. P. A. Bernstein and N. Goodman, “Power of Natural Semijoins,” SIAM J. Comput. 10, 751–771 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  35. O. A. Shcherbina, “Local Elimination Algorithms for Constraint Satisfaction Problems,” Tavricheskii Vestn. Inform. Mat., No. 1, 24–39 (2007).

  36. O. A. Shcherbina, “Treelike Decomposition and Discrete Optimization Problems (Survey),” Kibern. Sist. Anal., No. 4, 102–118 (2007).

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, University of Vienna, Vienna, Austria

    O. A. Shcherbina

Authors
  1. O. A. Shcherbina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. A. Shcherbina.

Additional information

Original Russian Text © O.A. Shcherbina, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 1, pp. 159–175.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shcherbina, O.A. Local elimination algorithms for solving sparse discrete problems. Comput. Math. and Math. Phys. 48, 152–167 (2008). https://doi.org/10.1134/S0965542508010120

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542508010120

Keywords

Search

Navigation