Abstract
We study the classical Bandwidth problem from the viewpoint of parameterized algorithms. In the Bandwidth problem we are given a graph G = (V,E) together with a positive integer k, and asked whether there is an bijective function β: {1, ..., n} →V such that for every edge uv ∈ E, |β − 1(u) − β − 1(v)| ≤ k. The problem is notoriously hard, and it is known to be NP-complete even on very restricted subclasses of trees. The best known algorithm for Bandwidth for small values of k is the celebrated algorithm by Saxe [SIAM Journal on Algebraic and Discrete Methods, 1980 ], which runs in time \(2^{{\mathcal{O}}(k)}n^{k+1}\). In a seminal paper, Bodlaender, Fellows and Hallet [STOC 1994 ] ruled out the existence of an algorithm with running time of the form \(f(k)n^{{\mathcal{O}}(1)}\) for any function f even for trees, unless the entire W-hierarchy collapses.
We initiate the search for classes of graphs where Bandwidth is fixed parameter tractable (FPT), that is, solvable in time \(f(k)n^{{\mathcal{O}}(1)}\) for some function f. In this paper we present an algorithm with running time \(2^{{\mathcal O}(k \log k)} n^2\) for Bandwidth on AT-free graphs, a well-studied graph class that contains interval, permutation, and cocomparability graphs. Our result is the first non-trivial FPT algorithm for Bandwidth on a graph class where the problem remains NP-complete.
This work is supported by the Research Council of Norway.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Assmann, S.F., Peck, G.W., Sysło, M.M., Zak, J.: The bandwidth of caterpillars with hairs of length 1 and 2. SIAM J. Alg. Disc. Meth. 2, 387–393 (1981)
Blache, G., Karpinski, M., Wirtgen, J.: On approximation intractability of the bandwidth problem. Technical report TR98-014, University of Bonn (1997)
Bodlaender, H.L., Fellows, M.R., Hallet, M.T.: Beyond NP-completeness for problems of bounded width (extended abstract): hardness for the W hierarchy. In: Proceedings of STOC 1994, pp. 449–458. ACM, New York (1994)
Brandstädt, A., Le, V.B., Spinrad, J.: Graph Classes: A Survey. SIAM, Philadelphia (1999)
Corneil, D.G., Olariu, S., Stewart, L.: Asteroidal Triple-Free Graphs. SIAM J. Disc. Math. 10, 399–430 (1997)
Cygan, M., Pilipczuk, M.: Exact and Approximate Bandwidth. In: Albers, S., et al. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 304–315. Springer, Heidelberg (2009)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)
Fishburn, P., Tanenbaum, P., Trenk, A.: Linear discrepancy and bandwidth. Order 18, 237–245 (2001)
Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
George, J.A., Liu, J.W.H.: Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, New Jersey (1981)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Heggernes, P., Kratsch, D., Meister, D.: Bandwidth of bipartite permutation graphs in polynomial time. Journal of Disc. Alg. (to appear)
Kaplan, H., Shamir, R.: Pathwidth, Bandwidth, and Completion Problems to Proper Interval Graphs with Small Cliques. SIAM J. Computing 25, 540–561 (1996)
Kleitman, D.J., Vohra, R.V.: Computing the bandwidth of interval graphs. SIAM J. Disc. Math. 3, 373–375 (1990)
Kloks, T., Kratsch, D., Le Borgne, Y., Müller, H.: Bandwidth of Split and Circular Permutation Graphs. In: Brandes, U., Wagner, D. (eds.) WG 2000. LNCS, vol. 1928, pp. 243–254. Springer, Heidelberg (2000)
Kloks, T., Kratsch, D., Müller, H.: Approximating the bandwidth for AT-free graphs. J. Alg. 32, 41–57 (1999)
Möhring, R.: Triangulating Graphs Without Asteroidal Triples. Discrete Applied Mathematics 64, 281–287 (1996)
Monien, B.: The Bandwidth-Minimization Problem for Caterpillars with Hair Length 3 is NP-Complete. SIAM J. Alg. Disc. Meth. 7, 505–512 (1986)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
Olariu, S.: An optimal greedy heuristic to color interval graphs. Information Processing Letters 37, 21–25 (1991)
Papadimitriou, C.: The NP-completeness of the bandwidth minimization problem. Computing 16, 263–270 (1976)
Saxe, J.B.: Dynamic-Programming Algorithms for Recognizing Small-Bandwidth Graphs in Polynomial Time. SIAM Journal on Algebraic and Discrete Methods 1, 363–369 (1980)
Yan, J.H.: The bandwidth problem in cographs. Tamsui Oxf. J. Math. Sci. 13, 31–36 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Golovach, P., Heggernes, P., Kratsch, D., Lokshtanov, D., Meister, D., Saurabh, S. (2009). Bandwidth on AT-Free Graphs. In: Dong, Y., Du, DZ., Ibarra, O. (eds) Algorithms and Computation. ISAAC 2009. Lecture Notes in Computer Science, vol 5878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10631-6_59
Download citation
DOI: https://doi.org/10.1007/978-3-642-10631-6_59
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-10630-9
Online ISBN: 978-3-642-10631-6
eBook Packages: Computer ScienceComputer Science (R0)