Abstract
We prove that a generalization of the edge dominating set problem, in which each edge e needs to be covered b e times for all e∈E, admits a linear time 2-approximation for general unweighted graphs and that it can be solved optimally for weighted trees. We show how to solve it optimally in linear time for unweighted trees and for weighted trees in which b e ∈{0,1} for all e∈E. Moreover, we show that it solves generalizations of weighted matching, vertex cover, and edge cover in trees.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger, A., Fukunaga, T., Nagamochi, H., Parekh, O.: Capacitated b-edge dominating set and related problems. Manuscript (2005)
Carr, R., Fujito, T., Konjevod, G., Parekh, O.: A 2 1/10-approximation algorithm for a generalization of the weighted edge-dominating set problem. J. Comb. Optim. 5, 317–326 (2001)
Fricke, G., Laskar, R.: Strong matchings on trees. In: Proceedings of the Twenty-third Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Boca Raton, FL, 1992, vol. 89, pp. 239–243 (1992)
Fujito, T.: On approximability of the independent/connected edge dominating set problems. Inf. Process. Lett. 79(6), 261–266 (2001)
Fujito, T., Nagamochi, H.: A 2-approximation algorithm for the minimum weight edge dominating set problem. Discrete Appl. Math. 118(3), 199–207 (2002)
Ghouila-Houri, A.: Caractérisation des matrices totalement unimodulaires. C. R. Acad. Sci. Paris 254, 1192–1194 (1962)
Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)
Hoffman, A.J., Kruskal, J.B.: Integral boundary points of convex polyhedra. In: Linear Inequalities and Related Systems. Annals of Mathematics Studies, vol. 38, pp. 223–246. Princeton University Press, Princeton (1956)
Horton, J.D., Kilakos, K.: Minimum edge dominating sets. SIAM J. Discrete Math. 6(3), 375–387 (1993)
Hwang, S.F., Chang, G.J.: The edge domination problem. Discuss. Math. Graph Theory 15(1), 51–57 (1995)
Mitchell, S.L., Hedetniemi, S.T.: Edge domination in trees. In: Proc. of the 8th Southeastern Conference on Combinatorics, Graph Theory and Computing, Louisiana State Univ., Baton Rouge, LA, 1977, pp. 489–509 (1977)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)
Parekh, O.: Edge dominating and hypomatchable sets. In: Proceedings of the 13th Annual ACM-SIAM Symposium On Discrete Mathematics (SODA-02), pp. 287–291. ACM Press, New York (2002)
Parekh, O.: Polyhedral techniques for graphic covering problems. Ph.D. thesis, Carnegie Mellon University (2002)
Parekh, O., Razouk, N.: A generalization of Gallai’s theorem. Manuscript (2005)
Preis, R.: Linear time \(\frac{1}{2}\) -approximation algorithm for maximum weighted matching in general graphs. In: STACS 99 (Trier). Lect. Notes in Comp. Sci., vol. 1563, pp. 259–269. Springer, Berlin (1999)
Schrijver, A.: Combinatorial Optimization. Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)
Srinivasan, A., Madhukar, K., Nagavamsi, P., Pandu Rangan, C., Chang, M.-S.: Edge domination on bipartite permutation graphs and cotriangulated graphs. Inf. Process. Lett. 56(3), 165–171 (1995)
Vazirani, V.: Approximation Algorithms. Springer, New York (2001)
Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38(3), 364–372 (1980)
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article is available at http://dx.doi.org/10.1007/s00453-011-9558-6.
Rights and permissions
About this article
Cite this article
Berger, A., Parekh, O. Linear Time Algorithms for Generalized Edge Dominating Set Problems. Algorithmica 50, 244–254 (2008). https://doi.org/10.1007/s00453-007-9057-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-007-9057-y