Skip to main content
SpringerLink
Log in

Space Saving by Dynamic Algebraization Based on Tree-Depth

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

Dynamic programming is widely used for exact computations based on tree decompositions of graphs. However, the space complexity is usually exponential in the treewidth. We study the problem of designing efficient dynamic programming algorithms based on tree decompositions in polynomial space. We show how to use a tree decomposition and extend the algebraic techniques of Lokshtanov and Nederlof (In: 42nd ACM Symposium on Theory of Computing, pp. 321–330, 2010) such that a typical dynamic programming algorithm runs in time O ∗(2h), where h is the tree-depth (Nešetřil et al., Eur. J. Comb. 27(6):1022–1041, 2006) of a graph. In general, we assume that a tree decomposition of depth h is given. We apply our algorithm to the problem of counting perfect matchings on grids and show that it outperforms other polynomial-space solutions. We also apply the algorithm to other set covering and partitioning 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

Fig. 1

Similar content being viewed by others

Notes

  1. O ∗ notation hides the polynomial factors of the expression.

References

  1. Amir, E.: Efficient Approximation for Triangulation of Minimum Treewidth Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence, pp 7–15 (2001)

    Google Scholar 

  2. Amir, E.: Approximation algorithms for treewidth. Algorithmica 56(4), 448–479 (2010). doi:10.1007/s00453-008-9180-4

    Article  MathSciNet  MATH  Google Scholar 

  3. Björklund, A.: Counting Perfect Matchings as Fast as Ryser 23Rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp 914–921 (2012)

    Chapter  Google Scholar 

  4. Björklund, A., Husfeldt, T.: Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier Meets Möbius: Fast Subset Convolution 39Th Annual ACM Symposium on Theory of Computing, pp 67–74 (2007)

    Google Scholar 

  6. Bodlaender, H.L.: Dynamic Programming on Graphs with Bounded Treewidth 15Th International Colloquium on Automata, Languages and Programming, pp 105–118 (1988)

    Chapter  Google Scholar 

  7. Bodlaender, H.L.: NC-Algorithms for Graphs with Small Treewidth 14Th International Workshop on Graph-Theoretic Concepts in Computer Science, pp 1–10 (1989)

    Google Scholar 

  8. Bodlaender, H.L.: A Linear Time Algorithm for Finding Tree-Decompositions of Small Treewidth 25Th Annual ACM Symposium on Theory of Computing, pp 226–234 (1993)

    Google Scholar 

  9. Bodlaender, H.L.: Discovering Treewidth 31St Conference on Current Trends in Theory and Practice of Computer Science, pp 1–16 (2005)

    Google Scholar 

  10. Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: An O(C k N) 5-Approximation Algorithm for Treewidth 54Th Annual IEEE Symposium on Foundations of Computer Science, pp 499–508 (2013)

    Google Scholar 

  11. Bodlaender, H.L., Gilbert, J.R., Kloks, T., Hafsteinsson, H.: Approximating Treewidth, Pathwidth, and Minimum Elimination Tree Height 17Th International Workshop on Graph-Theoretic Concepts in Computer Science, pp 1–12 (1992)

    Google Scholar 

  12. Bouchitté, V., Kratsch, D., Müller, H., Todinca, I.: On treewidth approximations. Discrete Appl. Math. 136(2-3), 183–196 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J. M.M., Wojtaszczyk, J.O.: Solving connectivity problems parameterized by treewidth in single exponential time 52nd Annual IEEE Symposium on Foundations of Computer Science, pp 150–159 (2011)

  14. Feige, U., Hajiaghayi, M., Lee, J.: Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput. 38(2), 629–657 (2008). doi:10.1137/05064299X

    Article  MathSciNet  MATH  Google Scholar 

  15. Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fürer, M.: Faster integer multiplication. SIAM J. Comput. 39(3), 979–1005 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gottlob, G., Leone, N., Scarcello, F.: Hypertree Decompositions: a Survey Mathematical Foundations of Computer Science, pp 37–57 (2001)

  18. Kenyon, C., Randall, D., Sinclair, A.: Approximating the number of monomer-dimer coverings of a lattice. J. Stat. Phys. 83(3), 637–659 (1996). doi:10.1007/BF02183743

  19. Kloks, T.: Treewidth, Computations and Approximations, vol. 842. Springer (1994)

  20. Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: A bound on the pathwidth of sparse graphs with applications to exact algorithms. SIAM J. Discret. Math. 23(1), 407–427 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Koutis, I.: Faster Algebraic Algorithms for Path and Packing Problems 35Th International Colloquium on Automata, Languages and Programming, pp 575–586 (2008)

  22. Koutis, I., Williams, R.: Limits and Applications of Group Algebras for Parameterized Problems 36Th International Colloquium on Automata, Languages and Programming, pp 653–664 (2009)

  23. Lokshtanov, D., Mnich, M., Saurabh, S.: Planar K-Path in Subexponential Time and Polynomial Space 37Th International Workshop on Graph-Theoretic Concepts in Computer Science, pp 262–270 (2011)

  24. Lokshtanov, D., Nederlof, J.: Saving Space by Algebraization. In: 42nd ACM Symposium on Theory of Computing, pp 321–330 (2010)

  25. Miller, G.L., Teng, S.H., Thurston, W., Vavasis, S.A.: Separators for sphere-packings and nearest neighbor graphs. J. ACM 44(1), 1–29 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nederlof, J.: Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica 65(4), 868–884 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nešetřil, J., de Mendez, P.O.: Tree-depth, subgraph coloring and homomorphism bounds. Eur. J. Comb. 27(6), 1022–1041 (2006)

  28. van Rooij, J.M.M., Bodlaender, H.L., Rossmanith, P.: Dynamic Programming on Tree Decompositions Using Generalised Fast Subset Convolution 17Th Annual European Symposium on Algorithms, pp 566–577 (2009)

  29. van Rooij, J.M.M., Nederlof, J., van Dijk, T.C.: Inclusion/Exclusion Meets Measure and Conquer 17Th Annual European Symposium on Algorithms, pp 554–565 (2009)

  30. Rota, G.C.: On the foundations of combinatorial theory. i. theory of möbius functions. Zeitschrift fü,r Wahrscheinlichkeitstheorie und Verwandte Gebiete 2(4), 340–368 (1964)

    Article  MATH  Google Scholar 

  31. Stanley, R., Rota, G.: Enumerative Combinatorics, vol. 1. Cambridge University Press (2000)

  32. Temperley, H., Fisher, M.: Dimer problem in statistical mechanics - an exact result. Philos. Mag. 6, 1061–1063 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  33. Woeginger, G.J.: Space and Time Complexity of Exact Algorithms: Some Open Problems (Invited Talk) 1St International Workshop on Parameterized and Exact Computation, pp 281–290 (2004)

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, The Pennsylvania State University, University Park, PA, USA

    Martin Fürer

  2. Google Incorporated, 1600 Amphitheatre Pkwy, Mountain View, CA, USA

    Huiwen Yu

  3. Pennsylvania State University, University Park, PA, USA

    Huiwen Yu

Authors
  1. Martin Fürer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huiwen Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Fürer.

Additional information

Research supported in part by NSF Grant CCF-0964655 and CCF-1320814

Part of this work has been done while the first author was visiting Theoretical Computer Science, ETH Zürich, Switzerland.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fürer, M., Yu, H. Space Saving by Dynamic Algebraization Based on Tree-Depth. Theory Comput Syst 61, 283–304 (2017). https://doi.org/10.1007/s00224-017-9751-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-017-9751-3

Keywords

Search

Navigation