Skip to main content

Polynomial Time and Parameterized Approximation Algorithms for Boxicity

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7535)

Abstract

The boxicity (cubicity) of a graph G, denoted by box(G) (respectively cub(G)), is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes (cubes) in ℝk. The problem of computing boxicity (cubicity) is known to be inapproximable in polynomial time even for graph classes like bipartite, co-bipartite and split graphs, within an O(n 0.5 − ε) factor for any ε > 0, unless NP = ZPP.

We prove that if a graph G on n vertices has a clique on n − k vertices, then box(G) can be computed in time \(n^2 2^{{O(k^2 \log k)}}\). Using this fact, various FPT approximation algorithms for boxicity are derived. The parameter used is the vertex (or edge) edit distance of the input graph from certain graph families of bounded boxicity - like interval graphs and planar graphs. Using the same fact, we also derive an \(O\left(\frac{n\sqrt{\log \log n}}{\sqrt{\log n}}\right)\) factor approximation algorithm for computing boxicity, which, to our knowledge, is the first o(n) factor approximation algorithm for the problem. We also present an FPT approximation algorithm for computing the cubicity of graphs, with vertex cover number as the parameter.

Keywords

  • Boxicity
  • Parameterized Algorithm
  • Approximation Algorithm

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

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   72.00
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adiga, A., Bhowmick, D., Chandran, L.S.: The hardness of approximating the boxicity, cubicity and threshold dimension of a graph. Discrete Appl. Math. 158, 1719–1726 (2010)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Adiga, A., Bhowmick, D., Chandran, L.S.: Boxicity and poset dimension. SIAM J. Discrete Math. 25(4), 1687–1698 (2011)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Adiga, A., Chitnis, R., Saurabh, S.: Parameterized Algorithms for Boxicity. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 366–377. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  4. Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)

    CrossRef  MATH  Google Scholar 

  6. Cai, L.: Parameterized complexity of vertex colouring. Discrete Applied Mathematics 127(3), 415–429 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Chandran, L.S., Sivadasan, N.: Boxicity and treewidth. J. Comb. Theory Ser. B 97, 733–744 (2007)

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Cozzens, M.B.: Higher and multi-dimensional analogues of interval graphs. Ph.D. thesis, Department of Mathematics, Rutgers University, New Brunswick, NJ (1981)

    Google Scholar 

  9. Fellows, M.R., Hermelin, D., Rosamond, F.A.: Well-Quasi-Orders in Subclasses of Bounded Treewidth Graphs. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 149–160. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  10. Grohe, M.: Computing crossing numbers in quadratic time. J. Comput. Syst. Sci. 68, 285–302 (2004)

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Appl. Math. 52(3), 233–252 (1994)

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Marx, D.: Parameterized coloring problems on chordal graphs. Theor. Comput. Sci. 351(3), 407–424 (2006)

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Marx, D., Schlotter, I.: Obtaining a planar graph by vertex deletion. Algorithmica 62(3-4), 807–822 (2012)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Niedermeier, R.: Invitation to fixed-parameter algorithms (2002)

    Google Scholar 

  15. Roberts, F.S.: On the boxicity and cubicity of a graph. In: Recent Progresses in Combinatorics, pp. 301–310. Academic Press, New York (1969)

    Google Scholar 

  16. Rosgen, B., Stewart, L.: Complexity results on graphs with few cliques. Discrete Mathematics and Theoretical Computer Science 9, 127–136 (2007)

    MathSciNet  MATH  Google Scholar 

  17. Thomassen, C.: Interval representations of planar graphs. J. Comb. Theory Ser. B 40, 9–20 (1986)

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Villanger, Y., Heggernes, P., Paul, C., Telle, J.A.: Interval completion is fixed parameter tractable. SIAM J. Comput. 38(5), 2007–2020 (2008)

    CrossRef  MathSciNet  Google Scholar 

  19. Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Alg. Disc. Meth. 3(3), 351–358 (1982)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Adiga, A., Babu, J., Chandran, L.S. (2012). Polynomial Time and Parameterized Approximation Algorithms for Boxicity. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-33293-7_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33292-0

  • Online ISBN: 978-3-642-33293-7

  • eBook Packages: Computer ScienceComputer Science (R0)