Advertisement

Algorithmica

, 56:129 | Cite as

Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

  • L. Sunil ChandranEmail author
  • Mathew C. Francis
  • Naveen Sivadasan
Article

Abstract

An axis-parallel k-dimensional box is a Cartesian product R 1×R 2×⋅⋅⋅×R k where R i (for 1≤ik) is a closed interval of the form [a i ,b i ] on the real line. For a graph G, its boxicity box (G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc.

A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a \(\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}\) approximation ratio for any constant c≥1 when d≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard.

We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in ⌈(Δ+2)ln n⌉ dimensions, where Δ is the maximum degree of G. This algorithm implies that box (G)≤⌈(Δ+2)ln n⌉ for any graph G. Our bound is tight up to a factor of ln n.

We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm.

Though our general upper bound is in terms of maximum degree Δ, we show that for almost all graphs on n vertices, their boxicity is O(d av ln n) where d av is the average degree.

Keywords

Boxicity Randomized algorithm Derandomization Random graph Intersection graphs 

References

  1. 1.
    Agarwal, P.K., van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom. Theory Appl. 11, 209–218 (1998) zbMATHGoogle Scholar
  2. 2.
    Angluin, D., Valiant, L.G.: Fast probabilistic algorithms for Hamiltonian circuits and matchings. J. Comput. Syst. Sci. 18, 155–193 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bellantoni, S., Hartman, I.B.-A., Przytycka, T., Whitesides, S.: Grid intersection graphs and boxicity. Discrete Math. 114, 41–49 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Berman, P., DasGupta, B., Muthukrishnan, S., Ramaswami, S.: Efficient approximation algorithms for tiling and packing problems with rectangles. J. Algorithms 41, 443–470 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybern. 11, 1–21 (1993) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bollobás, B.: Random Graphs, 2 edn. Cambridge University Press, Cambridge (2001) zbMATHGoogle Scholar
  7. 7.
    Chan, T.M.: Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms 46, 178–189 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chandran, L.S., Sivadasan, N.: Boxicity and Treewidth. J. Comb. Theory Ser. B 97(5), 733–744 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chandran, L.S., Francis, M.C., Sivadasan, N.: Boxicity and maximum degree. J. Comb. Theory Ser. B (2007). doi: 10.1016/j.jctb.2007.08.002 Google Scholar
  10. 10.
    Chandran, L.S., Francis, M.C., Sivadasan, N.: Geometric representation of graphs in low dimension using axis parallel boxes. Technical report available at http://arxiv.org/pdf/cs/0605013
  11. 11.
    Chang, Y.W., West, D.B.: Rectangle number for hyper cubes and complete multipartite graphs. In: 29th SE Conf. Comb., Graph Th. and Comp., Congr. Numer., vol. 132, pp. 19–28 (1998) Google Scholar
  12. 12.
    Chang, Y.W., West, D.B.: Interval number and boxicity of digraphs. In: Proceedings of the 8th International Graph Theory Conf. (1998) Google Scholar
  13. 13.
    Cozzens, M.B.: Higher and multidimensional analogues of interval graphs. Ph.D. thesis, Rutgers University, New Brunswick, NJ (1981) Google Scholar
  14. 14.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput. 34, 1302–1323 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Feinberg, R.B.: The circular dimension of a graph. Discrete Math. 25, 27–31 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hastad, J.: Clique is hard to approximate within n 1−ε. Acta Math. 182, 105–142 (1998) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kloks, T.: Treewidth: Computations and Approximations. Lecture Notes in Computer Science, vol. 842. Springer, Berlin (1994) zbMATHGoogle Scholar
  18. 18.
    Kratochvil, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Appl. Math. 52, 233–252 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Roberts, F.S.: On the boxicity and Cubicity of a graph. In: Recent Progresses in Combinatorics, pp. 301–310. Academic, New York (1969) Google Scholar
  20. 20.
    Rosgen, B., Stewart, L.: Complexity results on graphs with few cliques. Discrete Math. Theor. Comput. Sci. 9, 127–136 (2007) zbMATHMathSciNetGoogle Scholar
  21. 21.
    Scheinerman, E.R.: Intersection classes and multiple intersection parameters. Ph.D. thesis, Princeton University (1984) Google Scholar
  22. 22.
    Shearer, J.B.: A note on circular dimension. Discrete Math. 29, 103–103 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Thomassen, C.: Interval representations of planar graphs. J. Comb. Theory Ser. B 40, 9–20 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Trotter, W.T., West, J.D.B.: Poset boxicity of graphs. Discrete Math. 64, 105–107 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebraic Discrete Methods 3, 351–358 (1982) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • L. Sunil Chandran
    • 1
    Email author
  • Mathew C. Francis
    • 1
  • Naveen Sivadasan
    • 2
  1. 1.Dept. of Computer Science and AutomationIndian Institute of ScienceBangaloreIndia
  2. 2.Advanced Technology CenterTata Consultancy ServicesHyderabadIndia

Personalised recommendations