Elsevier

Discrete Applied Mathematics

Volume 159, Issue 17, 28 October 2011, Pages 2147-2164
Discrete Applied Mathematics

Exact algorithms for dominating set

https://doi.org/10.1016/j.dam.2011.07.001Get rights and content
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Abstract

The measure and conquer approach has proven to be a powerful tool to analyse exact algorithms for combinatorial problems like Dominating Set and Independent Set. This approach is used in this paper to obtain a faster exact algorithm for Dominating Set. We obtain this algorithm by considering a series of branch and reduce algorithms. This series is the result of an iterative process in which a mathematical analysis of an algorithm in the series with measure and conquer results in a convex or quasiconvex programming problem. The solution, by means of a computer, to this problem not only gives a bound on the running time of the algorithm, but can also give an indication on where to look for a new reduction rule, often giving a new, possibly faster algorithm. As a result, we obtain an O(1.4969n) time and polynomial space algorithm.

Highlights

► We give the currently fastest exact algorithm for the Dominating Set problem. ► We obtain this algorithm by considering a series of branch-and-reduce algorithms. ► Each algorithm in the series is analysed using the measure and conquer approach. ► This analysis is used to find new rules for the next algorithm in the series. ► The result is an O(1.4969n)-time polynomial-space algorithm.

Keywords

Exact algorithms
Exponential time algorithms
Branch and reduce
Measure and conquer
Dominating set
Computer aided algorithm design

Cited by (0)

This paper is the first full description of our work from which a preliminary version appeared at the 25th International Symposium on Theoretical Aspects of Computer Science (STACS2008) [25]. In this paper, all running times are improved compared to this preliminary version by choosing a slightly better measure, similar to [7]. This leads to new and improved running times for the Dominating Set problem. Also, for the first time, we give full proofs of the running times of the algorithms obtained in the presented improvement series.