Abstract
A k-dominating set of a graph, G, is a set of vertices, Dā V(G), such that for every vertex v ā V(G), either v ā D or there exists a vertex u ā D that is at distance at most k from v in G. We are interested in finding k-dominating sets of small cardinality. In this paper we consider a simple, yet efficient, randomised greedy algorithm for finding a small k-dominating set of regular graphs. We analyse the average-case performance of this heuristic by analysing its performance on random regular graphs using differential equations. This, in turn, proves an upper bound on the size of a minimum k-dominating set of random regular graphs.
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.
References
Chang, G.J. and Nemhauser, G.L.: The k-Domination and k-Stability Problems on Graphs. TR-540, School of Operations Research and Industrial Eng., Cornell University (1982)
Duckworth, W.: Greedy Algorithms and Cubic Graphs. PhD thesis, Department of Mathematics and Statistics, The University of Melbourne, Australia (2001)
Duckworth, W. and Wormald, N.C.: Minimum Independent Dominating Sets of Random Cubic Graphs. Random Structures and Algorithms. To Appear.
Favaron, O., Haynes, T.W. and Slater, P.J.: Distance-k Independent Domination Sequences. Journal of Combinatorial Mathematics and Combinatorial Computing (2000) 33 225ā237
Haynes, T.W., Hedetniemi, S.T. and Slater, P.J.: Domination in Graphs: Advanced topics. Marcel Dekker Inc. (1998) New York
Janson, S., FLuczak, T. and Rucinski, A.: Random Graphs. Wiley-Interscience (2000)
Johnson, D.S.: Approximation Algorithms for Combinatorial problems. In: Proceedings of the 5th Annual ACM STOC, Journal of Computer and System Sciences (1994) 9 256ā278
Kutten, S. and Peleg, D.: Fast Distributed Construction of Small k-dominating Sets and Applications. Journal of Algorithms (1998) 28(1) 40ā66
Molloy, M. and Reed, B.: The Dominating Number of a Random Cubic Graph. Random Structures and Algorithms (1995) 7(3) 209ā221
Raz, R. and Safra, S.: A Sub-Constant Error-Probability Low-Degree Test and a Sub-Constant Error-Probability PCP Characterization of NP. In: Proceedings of the 29th Annual ACM STOC (1999) 475ā484 (electronic)
Papadimitriou, C.H. and Yannakakis, M.: Optimization, Approximation and Complexity Classes. Journal of Computer and System Sciences (1991) 43(3) 425ā440
Reed, B.: Paths, Stars and the Number Three. Combinatorics, Probability and Computing (1996) 5 277ā295
Robinson, R.W. and Wormald, N.C.: Almost All Regular Graphs are Hamiltonian. Random Structures and Algorithms (1994) 5(2) 363ā374
Wormald, N.C.: Differential Equations for Random Processes and Random Graphs. Annals of Applied Probability (1995) 5 1217ā1235
Wormald, N.C.: Models of Random Regular Graphs. In: Surveys in Combinatorics (1999) 239ā298 Cambridge University Press
Wormald, N.C.: The Differential Equation Method for Random Graph Processes and Greedy Algorithms. In: Lectures on Approximation and Randomized Algorithms (1999) 73ā155, PWN, Warsaw
Zito, M.: Greedy Algorithms for Minimisation Problems in Random Regular Graphs. In: Proceedings of the 19th European Symposium on Algorithms. Lecture Notes in Computer Science (2001) 2161 524ā536, Springer-Verlag
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
Ā© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Duckworth, W., Mans, B. (2002). Small k-Dominating Sets of Regular Graphs. In: Rolim, J.D.P., Vadhan, S. (eds) Randomization and Approximation Techniques in Computer Science. RANDOM 2002. Lecture Notes in Computer Science, vol 2483. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45726-7_11
Download citation
DOI: https://doi.org/10.1007/3-540-45726-7_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44147-2
Online ISBN: 978-3-540-45726-8
eBook Packages: Springer Book Archive