Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T12:14:46.061Z Has data issue: false hasContentIssue false
  • English
  • Français

Solving the Minimum Independent Domination Set Problem inGraphs by Exact Algorithm and Greedy Heuristic∗

Published online by Cambridge University Press:  02 May 2013

Christian Laforest  and
Raksmey Phan
Christian Laforest
Affiliation:
LIMOS, CNRS UMR 6158, Université Blaise Pascal, Clermont–Ferrand Campus des Cézeaux, 24 avenue des Landais, 63173 Aubière Cedex, France.. [email protected].
Raksmey Phan
Affiliation:
LIMOS, CNRS UMR 6158, Université Blaise Pascal, Clermont–Ferrand Campus des Cézeaux, 24 avenue des Landais, 63173 Aubière Cedex, France.. [email protected].
Get access

Abstract

In this paper we present a new approach to solve the Minimum Independent Dominating Setproblem in general graphs which is one of the hardest optimization problem. We propose amethod using a clique partition of the graph, partition that can be obtained greedily. Weprovide conditions under which our method has a better complexity than the complexity ofthe previously known algorithms. Based on our theoretical method, we design in the secondpart of this paper an efficient algorithm by including cuts in the search process. We thenexperiment it and show that it is able to solve almost all instances up to 50 vertices inreasonable time and some instances up to several hundreds of vertices. To go further andto treat larger graphs, we analyze a greedy heuristic. We show that it often gives good(sometimes optimal) results in large instances up to 60   000 vertices in less than 20 s.That sort of heuristic is a good approach to get an initial solution for our exact method.We also describe and analyze some of its worst cases.

Keywords

Combinatorial optimizationheuristicsexact algorithmworst case analysisexperimentationsindependent dominating set in graphs
Type
Research Article
Information
RAIRO - Operations Research , Volume 47 , Issue 3 , July 2013 , pp. 199 - 221
Copyright
© EDP Sciences, ROADEF, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baez-Duarte, L., Hardy–ramanujan’s asymptotic formula for partitions and the central limit theorem. Adv. Math. 125 (1997) 114–120. Google Scholar
Bourgeois, N., Escoffier, B. and Paschos, V.T., Fast algorithm for min independent dominating set. SIROCCO, Lect. Notes Comput. Sci. 6058 (2010) 247–261. Google Scholar
M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP–Completeness (Series of Books in the Mathematical Sciences). W.H. Freeman and Co Ltd, first edition edition (1979).
Halldórsson, M.M.. Approximating the minimum maximal independence number. Inf. Process. Lett. 46 (1993) 169–172. Google Scholar
Harary, F. and Livingston, M., Independent domination in hypercubes. Appl. Math. Lett. 6 (1993) 27–28. Google Scholar
Haviland, J., Independent domination in triangle–free graphs. Discrete Math. 308 (2008) 3545–3550. Google Scholar
Johnson, D.S., Papadimitriou, C.H. and Yannakakis, M., On generating all maximal independent sets. Inf. Process. Lett. 27 (1988) 119–123. Google Scholar
F. Kuhn, T. Nieberg, T. Moscibroda and R. Wattenhofer, Local approximation schemes for ad hoc and sensor networks. DIALM-POMC (2005) 97–103.
J. Little, Branch and Bound Methods for Combinatorial Problems. Ulan Press (2012).
Liu, C. and Song, Y., Exact algorithms for finding the minimum independent dominating set in graphs. ISAAC, Lect. Notes Comput. Sci. 4288 (2006) 439–448. Google Scholar
Orlovich, Y.L., Gordon, V.S. and de Werra, D., On the inapproximability of independent domination in 2p3–free perfect graphs. Theor. Comput. Sci. 410 (2009) 977–982. Google Scholar
Potluri, A. and Negi, A., Some observations on algorithms for computing minimum independent dominating set. in Springer IC3, Commun. Comput. Inf. Sci. 168 (2011) 57–68. CrossRefGoogle Scholar
Shiu, W.C., Chen, X.-G. and Chan, W.H., Triangle–free graphs with large independent domination number. Discrete Optim. 7 (2010) 86–92. Google Scholar
Y. Song, T. Liu and K. Xu, Independent domination on tree convex bipartite graphs, in Frontiers in Algorithmics and Algorithmic Aspects in Information and Management, edited by J. Snoeyink, P. Lu, K. Su and L. Wang. Springer Berlin Heidelberg, Lect. Notes Comput. Sci. 7285 (2012) 129–138.
J. Steele, The Cauchy–Schwarz master class: an introduction to the art of mathematical inequalities. MAA problem books series. Cambridge University Press (2004).