Algorithmica

, Volume 77, Issue 3, pp 642–660

# Exact Algorithms for Minimum Weighted Dominating Induced Matching

• Min Chih Lin
• Michel J. Mizrahi
• Jayme L. Szwarcfiter
Article

## Abstract

Say that an edge of a graph G dominates itself and every other edge sharing a vertex of it. An edge dominating set of a graph $$G=(V,E)$$ is a subset of edges $$E' \subseteq E$$ which dominates all edges of G. In particular, if every edge of G is dominated by exactly one edge of $$E'$$ then $$E'$$ is a dominating induced matching. It is known that not every graph admits a dominating induced matching, while the problem to decide if it does admit it is NP-complete. In this paper we consider the problems of counting the number of dominating induced matchings and finding a minimum weighted dominating induced matching, if any, of a graph with weighted edges. We describe three exact algorithms for general graphs. The first runs in linear time for a given vertex dominating set of fixed size of the graph. The second runs in polynomial time if the graph admits a polynomial number of maximal independent sets. The third one is an $$O^*(1.1939^n)$$ time and polynomial (linear) space, which improves over the existing algorithms for exactly solving this problem in general graphs.

### Keywords

Exact algorithms Graph theory Dominating induced matchings

### References

1. 1.
Björklund, A.: Determinant sums for undirected hamiltonicity. In: Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, pp. 173–182 (Washington, DC, USA), FOCS ’10, IEEE Computer Society (2010)Google Scholar
2. 2.
Björklund, A., Husfeldt, T.: Exact graph coloring using inclusion-exclusion. In: Kao, M.-Y. (ed.) Encyclopedia of Algorithms. Springer, Berlin (2008)Google Scholar
3. 3.
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets mobius: fast subset convolution. In: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing (New York, NY, USA), STOC ’07, ACM, pp. 67–74 (2007)Google Scholar
4. 4.
Blank, M.: An estimate of the external stability of a graph without pendant vertices. Prikl. Math. Programmirovanie 10, 3–11 (1973)Google Scholar
5. 5.
Brandstädt, A., Hundt, C., Nevries, R.: Efficient edge domination on hole-free graphs in polynomial time. In: Proceedings of the 9th Latin American Conference on Theoretical Informatics, pp. 650–661 (Berlin, Heidelberg), LATIN’10, Springer (2010)Google Scholar
6. 6.
Brandstädt, A., Leitert, A., Rautenbach, D.: Efficient dominating and edge dominating sets for graphs and hypergraphs. In: (Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC, Lecture Notes in Computer Science, vol. 7676, pp. 267–277. Springer (2012)Google Scholar
7. 7.
Brandstädt, A., Lozin, V.V.: On the linear structure and clique-width of bipartite permutation graphs. Ars Comb. 67, 273–281 (2003)
8. 8.
Brandstädt, A., Mosca, R.: Dominating induced matchings for $$P_7$$-free graphs in linear time. In: Asano, T., Nakano, S.-I., Okamoto, Y., Watanabe, O. (eds.) ISAAC, Lecture Notes in Computer Science, vol. 7074, pp. 100–109. Springer (2011)Google Scholar
9. 9.
Cardoso, D.M., Korpelainen, N., Lozin, V.V.: On the complexity of the dominating induced matching problem in hereditary classes of graphs. Discrete Appl. Math. 159(7), 521–531 (2011)
10. 10.
Cardoso, D.M., Cerdeira, J.O., Delorme, C., Silva, P.C.: Efficient edge domination in regular graphs. Discrete Appl. Math. 156(15), 3060–3065 (2008)
11. 11.
Cardoso, D.M., Lozin, V.V.: Dominating induced matchings. In: Lipshteyn, M., Levit, V.E., McConnell, R.M. (eds.) Graph Theory, Computational Intelligence and Thought , Lecture Notes in Computer Science, vol. 5420, pp. 77–86. Springer (2009)Google Scholar
12. 12.
Cygan, M., Pilipczuk, M.: Even faster exact bandwidth. ACM Trans. Algorithms 8(1), 8 (2012)
13. 13.
Dahllöf, V., Jonsson, P.: An algorithm for counting maximum weighted independent sets and its applications. In: Eppstein, D. (ed.) SODA, pp. 292–298. ACM/SIAM, Philadelphia (2002)Google Scholar
14. 14.
Dahllöf, V., Jonsson, P., Wahlström, M.: Counting models for 2SAT and 3SAT formulae. Theor. Comput. Sci. 332(1–3), 265–291 (2005)
15. 15.
Fomin, F.V., Gaspers, S., Saurabh, S., Stepanov, A.A.: On two techniques of combining branching and treewidth. Algorithmica 54(2), 181–207 (2009)
16. 16.
Fomin, F.V., Kratsch, D.: Exact Exponential Algorithms. Texts in Theoretical Computer Science. Springer, Berlin (2010)
17. 17.
Grinstead, D.L., Slater, P.J., Sherwani, N.A., Holmes, N.D.: Efficient edge domination problems in graphs. Inf. Process. Lett. 48(5), 221–228 (1993)
18. 18.
Gupta, S., Raman, V., Saurabh, S.: Maximum r-regular induced subgraph problem: fast exponential algorithms and combinatorial bounds. SIAM J. Discrete Math. 26(4), 1758–1780 (2012)
19. 19.
Iwata, Y.: A faster algorithm for dominating set analyzed by the potential method. In: Marx, D., Rossmanith, P. (eds.) Parameterized and Exact Computation—6th International Symposium, IPEC 2011, Saarbrücken, Germany, September 6–8, 2011. Revised Selected Papers, Lecture Notes in Computer Science, vol. 7112, pp. 41–54. Springer (2011)Google Scholar
20. 20.
Korpelainen, N.: A polynomial-time algorithm for the dominating induced matching problem in the class of convex graphs. Electron. Notes Discrete Math. 32, 133–140 (2009)
21. 21.
Lin, M.C., Mizrahi, M.J., Szwarcfiter, J.L.: Exact algorithms for dominating induced matchings. CoRR abs/1301.7602 (2013)Google Scholar
22. 22.
Lin, M.C., Mizrahi, M.J., Szwarcfiter, J.L.: An O*(1.1939 n) time algorithm for minimum weighted dominating induced matching. In: Cai, L., Cheng, S.-W., Lam, T.W. (eds.) ISAAC, Lecture Notes in Computer Science, vol. 8283, pp. 558–567. Springer (2013)Google Scholar
23. 23.
Livingston, M., Stout, Q.F.: Distributing resources in hypercube computers. In: Proceedings of the Third Conference on Hypercube Concurrent Computers and Applications: Architecture, Software, Computer Systems, and General Issues, vol. 1, pp. 222–231 (New York, NY, USA), C3P, ACM (1988)Google Scholar
24. 24.
Lu, C.L., Ko, M.-T., Tang, C.Y.: Perfect edge domination and efficient edge domination in graphs. Discrete Appl. Math. 119(3), 227–250 (2002)
25. 25.
Lu, C.L., Tang, C.Y.: Solving the weighted efficient edge domination problem on bipartite permutation graphs. Discrete Appl. Math. 87(1–3), 203–211 (1998)
26. 26.
Milanic, M.: Hereditary efficiently dominatable graphs. J. Graph Theory 73(4), 400–424 (2013)
27. 27.
Ore, O.: Theory of graphs. Am. Math. Soc. Colloq. Publ. 38 (1962)Google Scholar
28. 28.
Reed, B.A.: Paths, stars and the number three. Comb. Probab. Comput. 5, 277–295 (1996)
29. 29.
Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6(3), 505–517 (1977)
30. 30.
van Rooij, J.M.M., Nederlof, J., van Dijk, T.C.: Inclusion/exclusion meets measure and conquer. In: Fiat, A., Sanders, P. (eds.) ESA, Lecture Notes in Computer Science, vol. 5757, pp. 554–565. Springer (2009)Google Scholar
31. 31.
Wahlström, M.: A tighter bound for counting max-weight solutions to 2sat instances. In: Proceedings of the 3rd International Conference on Parameterized and Exact Computation, pp. 202–213 (Berlin, Heidelberg). IWPEC’08, Springer (2008)Google Scholar
32. 32.
Gerhard, J.: Woeginger, Combinatorial optimization–Eureka, You Shrink!. Springer-Verlag New York Inc., New York (2003)Google Scholar
33. 33.
Xiao, M., Nagamochi, H.: Exact algorithms for dominating induced matching based on graph partition. Discrete Appl. Math. 190–191, 147–162 (2015)

## Copyright information

© Springer Science+Business Media New York 2015

## Authors and Affiliations

• Min Chih Lin
• 1
• 2
• Michel J. Mizrahi
• 1
• 2
• Jayme L. Szwarcfiter
• 3
• 4
1. 1.CONICET, Instituto de CálculoBuenos AiresArgentina
2. 2.Departamento de ComputaciónUniversidad de Buenos AiresBuenos AiresArgentina
3. 3.I. Mat., COPPE and NCE, Universidade Federal do Rio de JaneiroRio de JaneiroBrazil
4. 4.Instituto Nacional de MetrologiaQualidade e TecnologiaRio de JaneiroBrazil