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

Content Provider	Springer Nature Link
Author	Aazami, Ashkan
Copyright Year	2008
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.
Starting Page	429
Ending Page	456
Page Count	28
File Format	PDF
ISSN	13826905
Journal	Journal of Combinatorial Optimization
Volume Number	19
Issue Number	4
e-ISSN	15732886
Language	English
Publisher	Springer US
Publisher Date	2008-07-31
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Dominating set Power dominating set Planar graphs Approximation algorithms PTAS Hardness of approximation Tree-width Integer programming Operations Research/Decision Theory Optimization Theory of Computation Mathematical Modeling and Industrial Mathematics Convex and Discrete Geometry Combinatorics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Control and Optimization Discrete Mathematics and Combinatorics Computational Theory and Mathematics Computer Science Applications

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