Abstract
A caterpillar is a tree in which all vertices of degree three or more lie on one path, called the backbone. We present a polynomial time algorithm that produces a linear arrangement of the vertices of a caterpillar with bandwidth at most O(log n/loglog n) times the local density of the caterpillar, where the local density is a well known lower bound on the bandwidth. This result is best possible in the sense that there are caterpillars whose bandwidth is larger than their local density by a factor of Ω(log n/loglog n). The previous best approximation ratio for the bandwidth of caterpillars was O(log n). We show that any further improvement in the approximation ratio would require using linear arrangements that do not respect the order of the vertices of the backbone. We also show how to obtain a (1 + ε) approximation for the bandwidth of caterpillars in time \(2^{\tilde{O}(\sqrt{n/\epsilon})}\). This result generalizes to trees, planar graphs, and any family of graphs with treewidth \(\tilde{O}(\sqrt{n})\).
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References
Blache, G., Karpinski, M., Wirtgen, J.: On approximation intractability of the bandwidth problem. Technical report, University of Bonn (1997)
Chinn, P., Chvatálová, J., Dewdney, A., Gibbs, N.: The bandwidth problem for graphs and matrices - survey. Journal of Graph Theory 6(3), 223–254 (1982)
Chung, F.R., Seymour, P.D.: Graphs with small bandwidth and cutwidth. Discrete Mathematics 75, 113–119 (1989)
Dunagan, J., Vempala, S.: On euclidean embeddings and bandwidth minimization. In: Goemans, M.X., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) RANDOM 2001 and APPROX 2001. LNCS, vol. 2129, pp. 229–240. Springer, Heidelberg (2001)
Feige, U.: Approximating the bandwidth via volume respecting embeddings. J. Comput. Syst. Sci. 60(3), 510–539 (2000)
Feige, U.: Coping with the NP-hardness of the graph bandwidth problem. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, pp. 10–19. Springer, Heidelberg (2000)
Filmus, Y.: Master’s thesis, Weizmann Institute (2003)
Gupta: Improved bandwidth approximation for trees. In: Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms, pp. 788–793 (2000)
Haralambides, J., Makedon, F., Monien, B.: Bandwidth minimization: An approximation algorithm for caterpillars. Mathematical Systems Theory, 169–177 (1991)
Krauthgamer, R., Lee, J., Mendel, M., Naor, A.: Measured descent: A new embedding method for finite metrics. In: FOCS, pp. 434–443 (2004)
Monien, B.: The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete. SIAM J. Algebraic Discrete Methods 7(4), 505–512 (1986)
Papadimitriou, C.: The NP-completeness of the bandwidth minimization problem. Computing 16, 263–270 (1976)
Saxe, J.: Dynamic-programming algorithms for recognizing small-bandwidth graphs in polynomial time. SIAM J. Algebraic Discrete Methods 1, 363–369 (1980)
Unger, W.: The complexity of the approximation of the bandwidth problem. In: FOCS 1998: Proceedings of the 39th Annual Symposium on Foundations of Computer Science, pp. 82–91 (1998)
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Feige, U., Talwar, K. (2005). Approximating the Bandwidth of Caterpillars. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_6
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DOI: https://doi.org/10.1007/11538462_6
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