Abstract
In this paper we present a faster exact exponential time algorithm for the edge dominating set problem. Our algorithm uses O(1.3226n) time and polynomial space. The algorithm combines an enumeration approach based on enumerating minimal vertex covers with the branch and reduce paradigm. Its time bound is obtained using the measure and conquer technique. The algorithm is obtained by starting with a slower algorithm which is refined stepwise. In this way a series of algorithms appears, each one slightly faster than the previous, resulting in the O(1.3226n) time algorithm.
The techniques also gives faster exact algorithms for: minimum weight edge dominating set, minimum (weight) maximal matching, matrix domination and the parametrised version of minimum weight maximal matching.
This research was partially supported by project BRICKS (Basic Research for Creating the Knowledge Society).
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References
Carr, R., Fujito, T., Konjevod, G., Parekh, O.: A \(2\frac{1}{10}\) approximation algorithm for a generalization of the weighted edge-dominating set problem. Journal of Combinatorial Optimization 5, 317–326 (2001)
Downey, R.G., Fellows, M.R.: Fixed-parameter tractability and completeness. Congressus Numerantium 87, 161–178 (1992)
Eppstein, D.: Quasiconvex analysis of backtracking algorithms. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, pp. 781–790 (2004)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45, 634–652 (1998)
Fernau, H.: Edge dominating set: Efficient enumeration-based exact algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 140–151. Springer, Heidelberg (2006)
Fomin, F.V., Gaspers, S., Saurabh, S.: Branching and treewidth based exact algorithms. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 16–25. Springer, Heidelberg (2006)
Fomin, F.V., Grandoni, F., Kratsch, D.: Measure and conquer: Domination — a case study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 191–203. Springer, Heidelberg (2005)
Fomin, F.V., Grandoni, F., Kratsch, D.: Some new techniques in design and analysis of exact (exponential) algorithms. Bulletin of the EATCS 87, 47–77 (2005)
Harary, F.: Graph Theory. Addison-Wesley, Reading, MA (1969)
Held, M., Karp, R.: A dynamic programming approach to sequencing problems. J. SIAM 10, 196–210 (1962)
Iwama, K.: Worst-case upper bounds for kSAT. Bulletin of the EATCS 82, 61–71 (2004)
Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Information Processing Letters 27, 119–123 (1988)
Lawler, E.L.: A note on the complexity of the chromatic number problem. Information Processing Letters 5, 66–67 (1976)
Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Generating all maximal independent sets: NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9, 558–565 (1980)
Moon, J.W., Moser, L.: On cliques in graphs. Israel J. Math. 3, 23–28 (1965)
PlesnÃk, J.: Constrained weighted matchings and edge coverings in graphs. Disc. Appl. Math. 92, 229–241 (1999)
Raman, V., Saurabh, S., Sikdar, S.: Efficient exact algorithms through enumerating maximal independent sets and other techniques. Theory of Computing Systems 42, 563–587 (2007)
Randerath, B., Schiermeyer, I.: Exact algorithms for minimum dominating set. Technical Report zaik2005-501, Universität zu Köln, Cologne, Germany (2005)
Razgon, I.: Exact computation of maximum induced forest. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 160–171. Springer, Heidelberg (2006)
Robson, J.M.: Algorithms for maximum independent sets. J. Algorithms 7, 425–440 (1986)
Schöning, U.: Algorithmics in exponential time. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 36–43. Springer, Heidelberg (2005)
Tarjan, R.E., Trojanowski, A.: Finding a maximum independent set. SIAM J. Comput. 6, 537–546 (1977)
van Rooij, J.M.M., Bodlaender, H.L.: Exact algorithms for edge domination. Technical Report UU-CS-2007-051, Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands (2007)
van Rooij, J.M.M., Bodlaender, H.L.: Design by measure and conquer: A faster exact algorithm for dominating set. In: Proc. 24st Symp. Theoretical Aspects of Computer Science (STACS 2008) (2008)
Villanger, Y.: Improved exponential-time algorithms for treewidth and minimum fill-in. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 800–811. Springer, Heidelberg (2006)
Woeginger, G.J.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)
Woeginger, G.J.: Space and time complexity of exact algorithms: Some open problems (invited talk). In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 281–290. Springer, Heidelberg (2004)
Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38, 364–372 (1980)
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van Rooij, J.M.M., Bodlaender, H.L. (2008). Exact Algorithms for Edge Domination. In: Grohe, M., Niedermeier, R. (eds) Parameterized and Exact Computation. IWPEC 2008. Lecture Notes in Computer Science, vol 5018. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79723-4_20
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