Abstract
In the Imbalance Minimization problem we are given a graph G = (V,E) and an integer b and asked whether there is an ordering v 1 ...v n of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex v i is the absolute value of the difference between the number of neighbors to the left and right of v i . The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. We show that this problem is fixed parameter tractable and provide an algorithm that runs in time 2O(b logb) ·n O(1). This resolves an open problem of Kára et al. [COCOON 2005].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Amir, E.: Efficient approximation for triangulation of minimum treewidth. In: UAI, pp. 7–15 (2001)
Biedl, T.C., Chan, T.M., Ganjali, Y., Hajiaghayi, M.T., Wood, D.R.: Balanced vertex-orderings of graphs. Discrete Applied Mathematics 148(1), 27–48 (2005)
Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)
Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)
Downey, R.G., Fellows, M.R.: Parameterized complexity. Monographs in Computer Science. Springer, New York (1999)
Flum, J., Grohe, M.: Parameterized complexity theory. In: Texts in Theoretical Computer Science. An EATCS Series, Springer, Berlin (2006)
Gaspers, S., Messinger, M.-E., Nowakowski, R.J., Pralat, P.: Clean the graph before you draw it? Inf. Process. Lett. 109(10), 463–467 (2009)
Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)
Kant, G., He, X.: Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theor. Comput. Sci. 172(1-2), 175–193 (1997)
Kára, J., Kratochvíl, J., Wood, D.R.: On the complexity of the balanced vertex ordering problem. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 849–858. Springer, Heidelberg (2005)
Kára, J., Kratochvíl, J., Wood, D.R.: On the complexity of the balanced vertex ordering problem. Discrete Mathematics & Theoretical Computer Science 9(1) (2007)
Kinnersley, N.G.: The vertex separation number of a graph equals its path-width. Inf. Process. Lett. 42(6), 345–350 (1992)
Kirousis, L.M., Papadimitriou, C.H.: Interval graphs and searching. Discrete Mathematics 55(2), 181–184 (1985)
Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)
Papakostas, A., Tollis, I.G.: Algorithms for area-efficient orthogonal drawings. Comput. Geom. 9(1-2), 83–110 (1998)
Thilikos, D.M., Serna, M.J., Bodlaender, H.L.: Cutwidth i: A linear time fixed parameter algorithm. J. Algorithms 56(1), 1–24 (2005)
Thilikos, D.M., Serna, M.J., Bodlaender, H.L.: Cutwidth ii: Algorithms for partial w-trees of bounded degree. J. Algorithms 56(1), 25–49 (2005)
Wood, D.R.: Optimal three-dimensional orthogonal graph drawing in the general position model. Theor. Comput. Sci. 1-3(299), 151–178 (2003)
Wood, D.R.: Minimising the number of bends and volume in 3-dimensional orthogonal graph drawings with a diagonal vertex layout. Algorithmica 39(3), 235–253 (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lokshtanov, D., Misra, N., Saurabh, S. (2010). Imbalance Is Fixed Parameter Tractable. In: Thai, M.T., Sahni, S. (eds) Computing and Combinatorics. COCOON 2010. Lecture Notes in Computer Science, vol 6196. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14031-0_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-14031-0_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14030-3
Online ISBN: 978-3-642-14031-0
eBook Packages: Computer ScienceComputer Science (R0)