Document Open Access Logo

Exact and Approximate Digraph Bandwidth

Authors Pallavi Jain, Lawqueen Kanesh, William Lochet, Saket Saurabh, Roohani Sharma

Thumbnail PDF


  • Filesize: 0.74 MB
  • 15 pages

Document Identifiers

Author Details

Pallavi Jain
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Lawqueen Kanesh
  • The Institute of Mathematical Sciences, HBNI, India
William Lochet
  • University of Bergen, Norway
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, India
Roohani Sharma
  • The Institute of Mathematical Sciences, HBNI, India

Cite AsGet BibTex

Pallavi Jain, Lawqueen Kanesh, William Lochet, Saket Saurabh, and Roohani Sharma. Exact and Approximate Digraph Bandwidth. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 18:1-18:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


In this paper, we introduce a directed variant of the classical Bandwidth problem and study it from the view-point of moderately exponential time algorithms, both exactly and approximately. Motivated by the definitions of the directed variants of the classical Cutwidth and Pathwidth problems, we define Digraph Bandwidth as follows. Given a digraph D and an ordering sigma of its vertices, the digraph bandwidth of sigma with respect to D is equal to the maximum value of sigma(v)-sigma(u) over all arcs (u,v) of D going forward along sigma (that is, when sigma(u) < sigma (v)). The Digraph Bandwidth problem takes as input a digraph D and asks to output an ordering with the minimum digraph bandwidth. The undirected Bandwidth easily reduces to Digraph Bandwidth and thus, it immediately implies that Directed Bandwidth is {NP-hard}. While an O^*(n!) time algorithm for the problem is trivial, the goal of this paper is to design algorithms for Digraph Bandwidth which have running times of the form 2^O(n). In particular, we obtain the following results. Here, n and m denote the number of vertices and arcs of the input digraph D, respectively. - Digraph Bandwidth can be solved in O^*(3^n * 2^m) time. This result implies a 2^O(n) time algorithm on sparse graphs, such as graphs of bounded average degree. - Let G be the underlying undirected graph of the input digraph. If the treewidth of G is at most t, then Digraph Bandwidth can be solved in time O^*(2^(n + (t+2) log n)). This result implies a 2^(n+O(sqrt(n) log n)) algorithm for directed planar graphs and, in general, for the class of digraphs whose underlying undirected graph excludes some fixed graph H as a minor. - Digraph Bandwidth can be solved in min{O^*(4^n * b^n), O^*(4^n * 2^(b log b log n))} time, where b denotes the optimal digraph bandwidth of D. This allow us to deduce a 2^O(n) algorithm in many cases, for example when b <= n/(log^2n). - Finally, we give a (Single) Exponential Time Approximation Scheme for Digraph Bandwidth. In particular, we show that for any fixed real epsilon > 0, we can find an ordering whose digraph bandwidth is at most (1+epsilon) times the optimal digraph bandwidth, in time O^*(4^n * (ceil[4/epsilon])^n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Approximation algorithms
  • directed bandwidth
  • digraph bandwidth
  • approximation scheme
  • exact exponential algorithms


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. O. Amini, F.V. Fomin, and S. Saurabh. Counting Subgraphs via Homomorphisms. SIAM J. Discrete Math., 26(2):695-717, 2012. Google Scholar
  2. S.F. Assmann, G.W. Peck, M.M. Sysło, and J. Zak. The bandwidth of caterpillars with hairs of length 1 and 2. SIAM J. Alg. Discrete Meth., 2(4):387-393, 1981. Google Scholar
  3. G. Blache, M. Karpiński, and J. Wirtgen. On approximation intractability of the bandwidth problem. Citeseer, 1997. Google Scholar
  4. H.L. Bodlaender, M.R. Fellows, and M.T. Hallett. Beyond NP-completeness for problems of bounded width (extended abstract): hardness for the W hierarchy. In Proc. of STOC 1994, pages 449-458, 1994. Google Scholar
  5. M. Chudnovsky, A. Fradkin, and P. Seymour. Tournament Immersion and Cutwidth. J. Comb. Theory Ser. B, 102(1):93-101, 2012. Google Scholar
  6. M. Cygan and M. Pilipczuk. Faster Exact Bandwidth. In Proc. of WG 2008, pages 101-109, 2008. Google Scholar
  7. M. Cygan and M. Pilipczuk. Exact and approximate bandwidth. Theor. Comput. Sci., 411(40-42):3701-3713, 2010. Google Scholar
  8. M. Cygan and M. Pilipczuk. Bandwidth and distortion revisited. Discrete Appl. Math., 160(4-5):494-504, 2012. Google Scholar
  9. M. Cygan and M. Pilipczuk. Even Faster Exact Bandwidth. ACM Trans. Algorithms, 8(1):8:1-8:14, 2012. Google Scholar
  10. Marek Cygan and Marcin Pilipczuk. Faster exact bandwidth. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 101-109. Springer, 2008. Google Scholar
  11. J. Díaz, M. Serna, and D.M. Thilikos. Counting H-colorings of partial k-trees. Theor. Comput. Sci., 281(1-2):291-309, 2002. Google Scholar
  12. M.S. Dregi and D. Lokshtanov. Parameterized Complexity of Bandwidth on Trees. In Proc. of ICALP 2014, pages 405-416, 2014. Google Scholar
  13. U. Feige. Coping with the NP-hardness of the graph bandwidth problem. In Proc. of SWAT 2000, pages 10-19. Springer, Berlin, 2000. Google Scholar
  14. U. Feige and K. Talwar. Approximating the Bandwidth of Caterpillars. In Proc. of APPROX-RANDOM 2005, volume 3624, pages 62-73, 2005. Google Scholar
  15. M. Fürer, S. Gaspers, and S.P. Kasiviswanathan. An exponential time 2-approximation algorithm for bandwidth. Theor. Comput. Sci., 511:23-31, 2013. Google Scholar
  16. M. Garey, R. Graham, D. Johnson, and D. Knuth. Complexity Results for Bandwidth Minimization. SIAM J. Appl. Math., 34(3):477-495, 1978. Google Scholar
  17. M.R. Garey and D.S. Johnson. Computers and intractability, volume 29. wh freeman New York, 2002. Google Scholar
  18. P.A. Golovach, P. Heggernes, D. Kratsch, D. Lokshtanov, D. Meister, and S. Saurabh. Bandwidth on AT-free graphs. Theor. Comput. Sci., 412(50):7001-7008, 2011. Google Scholar
  19. P. Heggernes, D. Kratsch, and D. Meister. Bandwidth of bipartite permutation graphs in polynomial time. J. Discrete Algorithms, 7(4):533-544, 2009. Google Scholar
  20. D.J. Kleitman and R. Vohra. Computing the Bandwidth of Interval Graphs. SIAM J. Discrete Math., 3(3):373-375, 1990. Google Scholar
  21. R. Krauthgamer, J.R. Lee, M. Mendel, and A. Naor. Measured Descent: A New Embedding Method for Finite Metrics. In Proc. of FOCS 2004, pages 434-443, 2004. Google Scholar
  22. Yung-Ling Lai and Kenneth Williams. A survey of solved problems and applications on bandwidth, edgesum, and profile of graphs. Journal of graph theory, 31(2):75-94, 1999. Google Scholar
  23. Marek M. Karpinski, Jürgen Wirtgen, and A. Zelikovsky. An Approximation Algorithm for the Bandwidth Problem on Dense Graphs. Technical report, University of Bonn, 1997. Google Scholar
  24. B. Monien. The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete. SIAM J. Alg. Discrete Meth., 7(4):505-512, 1986. Google Scholar
  25. C.H. Papadimitriou. The NP-Completeness of the bandwidth minimization problem. Computing, 16(3):263-270, 1976. Google Scholar
  26. J.B. Saxe. Dynamic-programming algorithms for recognizing small-bandwidth graphs in polynomial time. SIAM J. Alg. Discrete Meth., 1(4):363-369, 1980. Google Scholar
  27. W. Unger. The complexity of the approximation of the bandwidth problem. In Proc. of FOCS 1998, pages 82-91, 1998. Google Scholar
  28. V. Vassilevska, R. Williams, and S.L.M. Woo. Confronting hardness using a hybrid approach. In Proc. of SODA, pages 1-10, 2006. Google Scholar
  29. J.H. Yan. The bandwidth problem in cographs. Tamsui Oxford J. Math. Sci, 13:31-36, 1997. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail