Abstract
Since in general it is NP-hard to solve the minimum dominating set problem even approximatively, a lot of work has been dedicated to central and distributed approximation algorithms on restricted graph classes. In this paper, we compromise between generality and efficiency by considering the problem on graphs of small arboricity a. These family includes, but is not limited to, graphs excluding fixed minors, such as planar graphs, graphs of (locally) bounded treewidth, or bounded genus. We give two viable distributed algorithms. Our first algorithm employs a forest decomposition, achieving a factor \({\mathcal{O}}(a^2)\) approximation in randomized time \({\mathcal{O}}(\log n)\). This algorithm can be transformed into a deterministic central routine computing a linear-time constant approximation on a graph of bounded arboricity, without a priori knowledge on a. The second algorithm exhibits an approximation ratio of \({\mathcal{O}}(a\log \Delta)\), where Δ is the maximum degree, but in turn is uniform and deterministic, and terminates after \({\mathcal{O}}(\log \Delta)\) rounds. A simple modification offers a trade-off between running time and approximation ratio, that is, for any parameter α ≥ 2, we can obtain an \({\mathcal{O}}(a \alpha \log_{\alpha} \Delta)\)-approximation within \({\mathcal{O}}(\log_{\alpha} \Delta)\) rounds.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Babai, L., Itai, A.: A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem. J. Algorithms 7(4), 567–583 (1986)
Baker, B.S.: Approximation Algorithms for NP-Complete Problems on Planar Graphs. J. ACM 41(1), 153–180 (1994)
Barenboim, L., Elkin, M.: Sublogarithmic Distributed MIS algorithm for Sparse Graphs using Nash-Williams Decomposition. Distributed Computing, 1–17 (2009)
Chlebk, M., Chlebkov, J.: Approximation Hardness of Dominating Set Problems in Bounded Degree Graphs. Information and Computation 206(11), 1264–1275 (2008)
Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit Disk Graphs. Discrete Math. 86(1-3), 165–177 (1990)
Czygrinow, A., Hańćkowiak, M., Wawrzyniak, W.: Fast Distributed Approximations in Planar Graphs. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 78–92. Springer, Heidelberg (2008)
Czygrinow, A., Hańćkowiak, M.: Distributed Almost Exact Approximations for Minor-Closed Families. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 244–255. Springer, Heidelberg (2006)
Czygrinow, A., Hanckowiak, M.: Distributed Approximation Algorithms for Weighted Problems in Minor-Closed Families. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 515–525. Springer, Heidelberg (2007)
Eppstein, D.: Diameter and Treewidth in Minor-Closed Graph Families. Algorithmica 27(3), 275–291 (2000)
Feige, U.: A Threshold of ln n for Approximating Set Cover. J. ACM 45(4), 634–652 (1998)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)
Grohe, M.: Local Tree-Width, Excluded Minors, and Approximation Algorithms. Combinatorica 23(4), 613–632 (2003)
Hunt, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs. Journal of Algorithms 26(2), 238–274 (1998)
Kuhn, F.: Personal Communication (2010)
Kuhn, F., Moscibroda, T., Wattenhofer, R.: What Cannot Be Computed Locally! In: Proc. 23rd Annual ACM Symposium on Principles of Distributed Computing, PODC (2004)
Kuhn, F., Moscibroda, T., Wattenhofer, R.: The Price of Being Near-Sighted. In: Proc. 17th ACM-SIAM Symposium on Discrete Algorithms, SODA (2006)
Kuhn, F., Wattenhofer, R.: Constant-Time Distributed Dominating Set Approximation. Distrib. Comput. 17(4), 303–310 (2005)
Lenzen, C., Oswald, Y.A., Wattenhofer, R.: What can be Approximated Locally? In: 20th ACM Symposium on Parallelism in Algorithms and Architecture, SPAA (June 2008)
Lenzen, C., Wattenhofer, R.: Leveraging Linial’s Locality Limit. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 394–407. Springer, Heidelberg (2008)
Linial, N.: Locality in Distributed Graph Algorithms. SIAM Journal on Computing 21(1), 193–201 (1992)
Luby, M.: A Simple Parallel Algorithm for the Maximal Independent Set Problem. SIAM J. Comput. 15(4), 1036–1055 (1986)
Métivier, Y., Robson, J.M., Saheb Djahromi, N., Zemmari, A.: An Optimal Bit Complexity Randomised Distributed MIS Algorithm. In: Proc. 16th International Colloquium on Structural Information and Communication Complexity. pp. 1–15. Piran Slovenia (2009)
Raz, R., Safra, S.: A Sub-Constant Error-Probability Low-Degree Test, and a Sub-Constant Error-Probability PCP Characterization of NP. In: Proc. of the 29th Annual ACM Symposium on Theory of Computing (STOC), pp. 475–484. ACM, New York (1997)
Schneider, J., Wattenhofer, R.: A Log-Star Distributed Maximal Independent Set Algorithm for Growth-Bounded Graphs. In: Proc. of the 27th Annual ACM Symposium on Principles of Distributed Computing, PODC (August 2008)
Takamizawa, K., Nishizeki, T., Saito, N.: Linear-Time Computability of Combinatorial Problems on Series-Parallel Graphs. J. ACM 29(3), 623–641 (1982)
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
Lenzen, C., Wattenhofer, R. (2010). Minimum Dominating Set Approximation in Graphs of Bounded Arboricity. In: Lynch, N.A., Shvartsman, A.A. (eds) Distributed Computing. DISC 2010. Lecture Notes in Computer Science, vol 6343. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15763-9_48
Download citation
DOI: https://doi.org/10.1007/978-3-642-15763-9_48
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15762-2
Online ISBN: 978-3-642-15763-9
eBook Packages: Computer ScienceComputer Science (R0)