Domination in graphs with bounded propagation: algorithms, formulations and hardness results

Ashkan Aazami ()
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Ashkan Aazami: University of Waterloo

Journal of Combinatorial Optimization, 2010, vol. 19, issue 4, No 1, 429-456

Abstract: Abstract We introduce a hierarchy of problems between the Dominating Set problem and the Power Dominating Set (PDS) problem called the ℓ-round power dominating set (ℓ-round PDS, for short) problem. For ℓ=1, this is the Dominating Set problem, and for ℓ≥n−1, this is the PDS problem; here n denotes the number of nodes in the input graph. In PDS the goal is to find a minimum size set of nodes S that power dominates all the nodes, where a node v is power dominated if (1) v is in S or it has a neighbor in S, or (2) v has a neighbor u such that u and all of its neighbors except v are power dominated. Note that rule (1) is the same as for the Dominating Set problem, and that rule (2) is a type of propagation rule that applies iteratively. The ℓ-round PDS problem has the same set of rules as PDS, except we apply rule (2) in “parallel” in at most ℓ−1 rounds. We prove that ℓ-round PDS cannot be approximated better than $2^{\log^{1-\epsilon}{n}}$ even for ℓ=4 in general graphs. We provide a dynamic programming algorithm to solve ℓ-round PDS optimally in polynomial time on graphs of bounded tree-width. We present a PTAS (polynomial time approximation scheme) for ℓ-round PDS on planar graphs for $\ell=O(\frac{\log{n}}{\log{\log{n}}})$ . Finally, we give integer programming formulations for ℓ-round PDS.

Keywords: Dominating set; Power dominating set; Planar graphs; Approximation algorithms; PTAS; Hardness of approximation; Tree-width; Integer programming (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (10)

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DOI: 10.1007/s10878-008-9176-7

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