Abstract
Inclusion/exclusion and measure and conquer are two central techniques from the field of exact exponential-time algorithms that recently received a lot of attention. In this paper, we show that both techniques can be used in a single algorithm. This is done by looking at the principle of inclusion/exclusion as a branching rule. This inclusion/exclusion-based branching rule can be combined in a branch-and-reduce algorithm with traditional branching rules and reduction rules. The resulting algorithms can be analysed using measure and conquer allowing us to obtain good upper bounds on their running times.
In this way, we obtain the currently fastest exact exponential-time algorithms for a number of domination problems in graphs. Among these are faster polynomial-space and exponential-space algorithms for #Dominating Set and Minimum Weight Dominating Set (for the case where the set of possible weight sums is polynomially bounded), and a faster polynomial-space algorithm for Domatic Number.
This approach is also extended in this paper to the setting where not all requirements in a problem need to be satisfied. This results in faster polynomial-space and exponential-space algorithms for Partial Dominating Set, and faster polynomial-space and exponential-space algorithms for the well-studied parameterised problem k-Set Splitting and its generalisation k-Not-All-Equal Satisfiability.
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Notes
An algorithm similar to Algorithm 4 has been published in an earlier draft of a part of this paper [69]. We note that the analysis published in [69] contains a mistake. We corrected this mistake and as a result, the preference order that can be found in the algorithm in [69] is no longer required. For the rest, both algorithms are very similar though they appear different. This is because Algorithm 4 is presented using red-blue dominating sets, while the Algorithm in [69] uses an equivalent presentation involving set covers. Furthermore, two reduction rules are omitted from Algorithm 4: one rule is superfluous (the base case in [69] is covered by the procedure CountRBDS2-DP(G,m)), and the other rule (connected components) is never used in the analysis and could thus be omitted.
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Acknowledgements
We thank Hans L. Bodlaender for his guidance and enthusiasm during this research.
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Preliminary parts of this paper have appeared in the conference proceedings of three different conferences. Preliminary parts appeared under the title ‘Inclusion/Exclusion Meets Measure and Conquer: Exact algorithms for counting dominating sets’ at the 17th Annual European Symposium on Algorithms (ESA 2009), Lecture Notes in Computer Science 5757, pp. 554–565, under the title ‘Polynomial Space Algorithms for Counting Dominating Sets and the Domatic Number’ at the 7th International Conference on Algorithms and Complexity (CIAC 2010), Lecture Notes in Computer Science 6078, pp. 73–84, and under the title ‘Inclusion/Exclusion Branching for Partial Dominating Set and Set Splitting’ at the 5th International Symposium on Parameterized and Exact Computation (IPEC 2011), Lecture Notes in Computer Science 6478, pp. 204–215. The last paper received the best student paper award of IPEC 2011.
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Nederlof, J., van Rooij, J.M.M. & van Dijk, T.C. Inclusion/Exclusion Meets Measure and Conquer. Algorithmica 69, 685–740 (2014). https://doi.org/10.1007/s00453-013-9759-2
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DOI: https://doi.org/10.1007/s00453-013-9759-2