Abstract
We show how to count all minimum weighted dominating sets of a graph on n vertices in time . Our algorithm is a combination of branch and bound approach along with dynamic programming on graphs with bounded treewidth. To achieve bound we introduce a technique of measuring running time of our algorithm by combining measure and conquer approach with linear programming.
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
Angelsmark, O., Jonsson, P.: Improved algorithms for counting solutions in constraint satisfaction problems. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 81–95. Springer, Heidelberg (2003)
Bax, E.T., Franklin, J.A: finite-difference sieve to count paths and cycles by length. Inf. Process. Lett. 60(4), 171–176 (1996)
Björklund, A., Husfeldt, T.: Inclusion-exclusion algorithms for counting set partitions. In: FOCS 2006. Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 575–582. IEEE Computer Society Press, Los Alamitos (2006)
Dahllöf, V., Jonsson, P.: An algorithm for counting maximum weighted independent sets and its applications. In: SODA 2002. Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, pp. 292–298. ACM Press, New York (2002)
Dahllöf, V., Jonsson, P., Wahlström, M.: Counting models for 2 SAT and 3 SAT formulae. Theoretical Computer Science 332 332(1-3), 265–291 (2005)
Fomin, F.V., Gaspers, S., Saurabh, S.: Branching and treewidth based exact algorithms. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 16–25. Springer, Heidelberg (2005)
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)
Fomin, F.V., Grandoni, F., Pyatkin, A.V., Stepanov, A.A.: Bounding the number of minimal dominating sets: a measure and conquer approach. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 573–582. Springer, Heidelberg (2005)
Fomin, F.V., Kratsch, D., Woeginger, G.J.: Exact (exponential) algorithms for the dominating set problem. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 245–256. Springer, Heidelberg (2004)
Fürer, M., Kasiviswanathan, S.P.: Algorithms for counting 2-SAT solutions and colorings with applications. Electronic Colloquium on Computational Complexity (ECCC) 33 (2005)
Grandoni, F.: A note on the complexity of minimum dominating set. Journal of Discrete Algorithms 4(2), 209–214 (2006)
Haynes, T.W., Hedetniemi, S.T.: Domination in graphs (Advanced topics). Monographs and Textbooks in Pure and Applied Mathematics, vol. 209. Marcel Dekker Inc., New York (1998)
Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Stearns, R.E.: The complexity of planar counting problems. SIAM Journal on Computing 27(4), 1142–1167 (1998)
Jerrum, M.: Counting, sampling and integrating: algorithms and complexity. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2003)
Karp, R.M.: Dynamic programming meets the principle of inclusion and exclusion. Operations Research Letters 1 2(1981/82), 49–51 (1981)
Koivisto, M.: An O(2n) algorithm for graph coloring and other partitioning problems via inclusion-exclusion. In: FOCS 2006. Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 583–590. IEEE Computer Society Press, Los Alamitos (2006)
Mölle, D., Richter, S., Rossmanith, P.: Enumerate and expand: Improved algorithms for connected vertex cover and tree cover. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 270–280. Springer, Heidelberg (2006)
Randerath, B., Schiermeyer, I.: Exact algorithms for MINIMUM DOMINATING SET. Technical Report zaik-469, Zentrum für Angewandte Informatik Köln, Germany (2004)
Ryser, H.J.: Combinatorial mathematics. The Carus Mathematical Monographs, No. 14. Published by The Mathematical Association of America (1963)
Schöning, U.: Algorithmics in exponential time. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 36–43. Springer, Heidelberg (2005)
Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)
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)
Zhang, W.: Number of models and satisfiability of sets of clauses. Theoretical Computer Science 155(1), 277–288 (1996)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fomin, F.V., Stepanov, A.A. (2007). Counting Minimum Weighted Dominating Sets. In: Lin, G. (eds) Computing and Combinatorics. COCOON 2007. Lecture Notes in Computer Science, vol 4598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73545-8_18
Download citation
DOI: https://doi.org/10.1007/978-3-540-73545-8_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73544-1
Online ISBN: 978-3-540-73545-8
eBook Packages: Computer ScienceComputer Science (R0)