Abstract
The lower and the upper irredundance numbers of a graph G, denoted ir(G) and IR(G) respectively, are conceptually linked to domination and independence numbers and have numerous relations to other graph parameters. It is a long-standing open question whether determining these numbers for a graph G on n vertices admits exact algorithms running in time less than the trivial Ω(2n) enumeration barrier. We solve this open problem by devising parameterized algorithms for the duals of the natural parameterizations of the problems with running times faster than \(\mathcal{O}^*(4^{k})\). For example, we present an algorithm running in time \(\mathcal{O}^*(3.069^{k}))\) for determining whether IR(G) is at least n − k. Although the corresponding problem has been shown to be in FPT by kernelization techniques, this paper offers the first parameterized algorithms with an exponential dependency on the parameter in the running time. Furthermore, these seem to be the first examples of a parameterized approach leading to a solution to a problem in exponential time algorithmics where the natural interpretation as exact exponential-time algorithms fails.
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
Favaron, O., Haynes, T.W., Hedetniemi, S.T., Henning, M.A., Knisley, D.J.: Total irredundance in graphs. Discrete Mathematics 256(1-2), 115–127 (2002)
Allan, R.B., Laskar, R.: On domination and independent domination numbers of a graph. Discrete Mathematics 23(2), 73–76 (1978)
Favaron, O.: Two relations between the parameters of independence and irredundance. Discrete Mathematics 70(1), 17–20 (1988)
Fellows, M.R., Fricke, G., Hedetniemi, S.T., Jacobs, D.P.: The private neighbor cube. SIAM J. Discrete Math. 7(1), 41–47 (1994)
Hedetniemi, S.T., Laskar, R., Pfaff, J.: Irredundance in graphs: a survey. Congr. Numer. 48, 183–193 (1985)
Bollobás, B., Cockayne, E.J.: Graph-theoretic parameters concerning domination, independence, and irredundance. J. Graph Theory 3, 241–250 (1979)
Cockayne, E.J., Grobler, P.J.P., Hedetniemi, S.T., McRae, A.A.: What makes an irredundant set maximal? J. Combin. Math. Combin. Comput. 25, 213–224 (1997)
Chang, M.S., Nagavamsi, P., Rangan, C.P.: Weighted irredundance of interval graphs. Information Processing Letters 66, 65–70 (1998)
Cockayne, E.J., Hedetniemi, S.T., Miller, D.J.: Properties of hereditary hypergraphs and middle graphs. Canad. Math. Bull. 21(4), 461–468 (1978)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 208. Marcel Dekker, New York (1998)
Colbourn, C.J., Proskurowski, A.: Concurrent transmissions in broadcast networks. In: Paredaens, J. (ed.) ICALP 1984. LNCS, vol. 172, pp. 128–136. Springer, Heidelberg (1984)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theoretical Computer Science 351(3), 446–458 (2006)
Downey, R.G., Fellows, M.R., Raman, V.: The complexity of irredundant sets parameterized by size. Discrete Applied Mathematics 100, 155–167 (2000)
Fomin, F.V., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. Journal of the ACM 56(5) (2009)
Fomin, F.V., Iwama, K., Kratsch, D., Kaski, P., Koivisto, M., Kowalik, L., Okamoto, Y., van Rooij, J., Williams, R.: 08431 Open problems – Moderately exponential time algorithms. In: Moderately Exponential Time Algorithms. Dagstuhl Seminar Proceedings, vol. 08431 (2008)
Cygan, M., Pilipczuk, M., Wojtaszczyk, J.O.: Irredundant set faster than O(2n). In: These proceedings
Chor, B., Fellows, M., Juedes, D.: Linear kernels in linear time, or how to save k colors in O(n 2) steps. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 257–269. Springer, Heidelberg (2004)
Fellows, M.R.: Blow-ups, win/win’s, and crown rules: Some new directions in FPT. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 1–12. Springer, Heidelberg (2003)
Telle, J.A.: Complexity of domination-type problems in graphs. Nordic. J. of Comp. 1, 157–171 (1994)
Telle, J.A.: Vertex Partitioning Problems: Characterization, Complexity and Algorithms on Partial k-Trees. PhD thesis, Department of Computer Science, University of Oregon, USA (1994)
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
Binkele-Raible, D. et al. (2010). A Parameterized Route to Exact Puzzles: Breaking the 2n-Barrier for Irredundance. In: Calamoneri, T., Diaz, J. (eds) Algorithms and Complexity. CIAC 2010. Lecture Notes in Computer Science, vol 6078. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13073-1_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-13073-1_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13072-4
Online ISBN: 978-3-642-13073-1
eBook Packages: Computer ScienceComputer Science (R0)