An improved exact algorithm for the domatic number problem

Riege, Tobias; Rothe, Jörg; Spakowski, Holger; Yamamoto, Masaki

doi:10.1016/j.ipl.2006.08.010

Cited by 11 publications

(1 citation statement)

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“…For example, using this kind of approach and the same measure which is used here for MDS, Fomin et al [2008a] proved that the number of minimal dominating sets in a graph is O * (2 0.783 n ). Based on this result, they also derived the first non-trivial exact algorithms for the domatic number problem and for the minimumweight dominating set problem (see also Björklund and Husfeldt [2006], Fomin and Stepanov [2007], Koivisto [2006], and Riege et al [2007]). The bounds on the number of minimal feedback vertex sets (or maximal induced forests) obtained in Fomin et al [2008b] are also based on Measure & Conquer.…”

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confidence: 99%

A measure & conquer approach for the analysis of exact algorithms

Fomin

Grandoni

Kratsch³

2009

J. ACM

215

143

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For more than 40 years, Branch & Reduce exponential-time backtracking algorithms have been among the most common tools used for finding exact solutions of NP-hard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worst-case running time bounds. Motivated by this, we use an approach, that we call "Measure & Conquer", as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worst-case time analysis.In order to show the potentialities of Measure & Conquer, we consider two well-studied NP-hard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is Preliminary parts of this article appeared in competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis).Our examples show that a good choice of the measure, made in the very first stages of exact algorithms design, can have a tremendous impact on the running time bounds achievable.

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confidence: 99%