Abstract
Several NP-hard problems, like Maximum Independent Set, Coloring, and Max-Cut are polynomial time solvable on bipartite graphs. An equivalent characterization of bipartite graphs is that it is the set of all graphs that do not contain any odd length cycle. Thus, a natural question here is what happens to the complexity of these problems if we know that the length of the longest odd cycle is bounded by k? Let \({\mathcal O}_k\) denote the set of all graphs G such that the length of the longest odd cycle is upper bounded by k. Hsu, Ikura and Nemhauser [Math. Programming, 1981] studied the effect of avoiding long odd cycle for the Maximum Independent Set problem and showed that a maximum sized independent set on a graph \(G\in{\mathcal O}_k\) on n vertices can be found in time n O(k). Later, Grötschel and Nemhauser [Math. Programming, 1984] did a similar study for Max-Cut and obtained an algorithm with running time n O(k) on a graph \(G\in{\mathcal O}_k\) on n vertices.
In this paper, we revisit these problems together with q -Coloring and observe that all of these problems admit algorithms with running time O(c k n O(1)) on a graph \(G\in{\mathcal O}_k\) on n vertices. Thus, showing that all these problems are fixed parameter tractable when parameterized by the length of the longest odd cycle of the input graph. However, following the recent trend in parameterized complexity, we also study the kernelization complexity of these problems. We show that Maximum Independent Set, q -Coloring for some fixed q ≥ 3 and Max-Cut do not admit a polynomial kernel unless \(\mbox{\sc co-NP} \subseteq \mbox{\sc NP}/\mbox{\textrm{poly}}\), when parameterized by k, the length of the longest odd cycle.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms. Addison-Wesley (1974)
Alon, N., Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: Solving MAX-r-SAT above a tight lower bound. In: SODA, pp. 511–517 (2010)
Birmele, E.: Tree-width and circumference of graphs. Journal of Graph Theory 43(1), 24–25 (2003)
Bodlaender, H.L.: Kernelization: New Upper and Lower Bound Techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)
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., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (meta) kernelization. In: FOCS, pp. 629–638 (2009)
Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Preprocessing for Treewidth: A Combinatorial Analysis through Kernelization. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 437–448. Springer, Heidelberg (2011)
Bodlaender, H.L., Thomassé, S., Yeo, A.: Analysis of data reduction: Transformations give evidence for non-existence of polynomial kernels. Technical report (2008)
Dell, H., van Melkebeek, D.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. In: STOC, pp. 251–260 (2010)
Downey, R.G., Fellows, M.R.: Parameterized Complexity, 530 p. Springer (1999)
Flum, J., Grohe, M.: Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series). Springer-Verlag New York, Inc., Secaucus (2006)
Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: SODA, pp. 503–510 (2010)
Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: STOC, pp. 133–142 (2008)
Grötschel, M., Nemhauser, G.L.: A polynomial algorithm for the max-cut problem on graphs without long odd cycles. Math. Programming 29(1), 28–40 (1984)
Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)
Hsu, W.L., Ikura, Y., Nemhauser, G.L.: A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles. Math. Programming 20(2), 225–232 (1981)
Jansen, B.M.P., Bodlaender, H.L.: Vertex cover kernelization revisited: Upper and lower bounds for a refined parameter. In: STACS, pp. 177–188 (2011)
Jansen, B.M.P., Kratsch, S.: Data Reduction for Graph Coloring Problems. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 90–101. Springer, Heidelberg (2011)
Jansen, B.M.P., Kratsch, S.: On Polynomial Kernels for Structural Parameterizations of Odd Cycle Transversal. In: Marx, D., Rossmanith, P. (eds.) IPEC 2011. LNCS, vol. 7112, pp. 132–144. Springer, Heidelberg (2012)
Kratsch, S., Wahlström, M.: Compression via matroids: a randomized polynomial kernel for odd cycle transversal. In: SODA, pp. 94–103 (2012)
Niedermeier, R.: Invitation to Fixed Parameter Algorithms (Oxford Lecture Series in Mathematics and Its Applications). Oxford University Press, USA (2006)
Raman, V., Saurabh, S.: Short cycles make W-hard problems hard: FPT algorithms for W-hard problems in graphs with no short cycles. Algorithmica 52(2), 203–225 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Panolan, F., Rai, A. (2012). On the Kernelization Complexity of Problems on Graphs without Long Odd Cycles. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_38
Download citation
DOI: https://doi.org/10.1007/978-3-642-32241-9_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-32240-2
Online ISBN: 978-3-642-32241-9
eBook Packages: Computer ScienceComputer Science (R0)