Abstract
A linear arrangement (LA) is an assignment of distinct integers to the vertices of a graph. The cost of an LA is the sum of lengths of the edges of the graph, where the length of an edge is defined as the absolute value of the difference of the integers assigned to its ends. For many application one hopes to find an LA with small cost. However, it is a classical NP-complete problem to decide whether a given graph G admits an LA of cost bounded by a given integer. Since every edge of G contributes at least one to the cost of any LA, the problem becomes trivially fixed-parameter tractable (FPT) if parameterized by the upper bound of the cost. Fernau asked whether the problem remains FPT if parameterized by the upper bound of the cost minus the number of edges of the given graph; thus whether the problem is FPT “parameterized above guaranteed value.” We answer this question positively by deriving an algorithm which decides in time O(m + n + 5.88k) whether a given graph with m edges and n vertices admits an LA of cost at most m + k (the algorithm computes such an LA if it exists). Our algorithm is based on a procedure which generates a problem kernel of linear size in linear time for a connected graph G. We also prove that more general parameterized LA problems stated by Serna and Thilikos are not FPT, unless P = NP.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Cesati, M.: Compendium of parameterized problems (September 2005), http://bravo.ce.uniroma2.it/home/cesati/research/compendium.pdf
Chung, F.R.K.: On optimal linear arrangements of trees. Comp. & Maths. with Appls. 10, 43–60 (1984)
Diestel, R.: Graph Theory, 2nd edn. Springer, New York (2000)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)
Estivill-Castro, V., Fellows, M.R., Langston, M.A., Rosamond, F.A.: FPT is P-Time extremal structure I. In: Broersma, H., Johnson, M., Szeider, S. (eds.) Algorithms and Complexity in Durham 2005, Proceedings of the first ACiD Workshop. Texts in Algorithmics, vol. 4, pp. 1–41. King’s College Publications (2005)
Fernau, H.: Parameterized Algorithmics: A Graph-theoretic Approach. Habilitation thesis, U. Tübingen (2005)
Fernau, H.: Parameterized Algorithmics for Linear Arrangement Problems. Talk at Dagstuhl (July 2005), slides at: http://www.dagstuhl.de/files/Materials/05/05301/05301.FernauHenning.Slides.pdf
Fernau, H.: Parameterized Algorithmics for Linear Arrangement Problems (manscript) (July 2005), http://homepages.feis.herts.ac.uk/~comrhf/papers/ola.pdf
Flum, J., Grohe, M.: Describing parameterized complexity classes. Information and Computation 187, 291–319 (2003)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Garey, M.R., Johnson, D.R.: Computers and Intractability. W.H. Freeman & Comp., New York (1979)
Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1, 237–267 (1976)
Goldberg, M.K., Klipker, I.A.: Minimal placing pf trees on a line. Tech. Report, Physico-Technical Institute of Low Temperatures, Ukranian SSR Acad. of Sciences, USSR (1976) (in Russian)
Harper, L.H.: Optimal assignments of numbers to vertices. J. Soc. Indust. Appl. Math. 12, 131–135 (1964)
Mahajan, M., Raman, V.: Parameterizing above guaranteed values: MaxSat and MaxCut. J. Algorithms 31, 335–354 (1999)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (forthcoming, 2006)
Shiloach, Y.: A minimum linear arrangement algorithm for undirected trees. SIAM J. Comp. 8, 15–32 (1979)
Serna, M., Thilikos, D.M.: Parameterized complexity for graph layout problems. EATCS Bulletin 86, 41–65 (2005)
Tarjan, R.E.: Depth first search and linear graph algorithms. SIAM J. Comput. 1, 146–160 (1972)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gutin, G., Rafiey, A., Szeider, S., Yeo, A. (2006). The Linear Arrangement Problem Parameterized Above Guaranteed Value. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds) Algorithms and Complexity. CIAC 2006. Lecture Notes in Computer Science, vol 3998. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11758471_34
Download citation
DOI: https://doi.org/10.1007/11758471_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34375-2
Online ISBN: 978-3-540-34378-3
eBook Packages: Computer ScienceComputer Science (R0)