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On Approximating (Connected) 2-Edge Dominating Set by a Tree

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Abstract

The edge dominating set problem (EDS) is to compute a minimum size edge set such that every edge is dominated by some edge in it. This paper considers a variant of EDS with extensions of multiple and connected dominations combined. In the b-EDS problem, each edge needs to be dominated b times. Connected EDS requires an edge dominating set to be connected while it has to form a tree in Tree Cover. Although each of EDS, b-EDS, and Connected EDS (or Tree Cover) has been well studied, each known to be approximable within 2 (or 8/3 for b-EDS in general), nothing is known when these extensions are imposed simultaneously on EDS unlike in the case of the (vertex) dominating set problem. We consider Connected 2-EDS and 2-Tree Cover (i.e., a combination of 2-EDS and Tree Cover), and present a polynomial algorithm approximating each within 2. Moreover, it will be shown that the single tree computed is no larger than twice the optimum for (not necessarily connected) 2-EDS, thus also approximating 2-EDS equally well. It also implies that 2-EDS with clustering properties can be approximated within 2 as well.

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Notes

  1. Consider for instance a star S n with the center node u having n leaves, and compose a graph G by attaching a distinct triangle to each leaf of S n . Then, a connected 2-eds needs to include all the edges of S n and at least 2 edges from each triangle, totaling to 3n edges. A maximum matching on the other hand is of size n + 1, consisting of one edge (u, v) in S n , the edge not incident to v from the triangle attached to v, and one edge from each of the remaining triangles.

  2. Think of the case of H ≃ K s, s+1 for instance, which can appear as a subgraph in \(G[A\cup D]\). Since any basic tree T spanning H consists of s many 2-edges, |E(T)| = 2s and hence, (|E(T)| + 1)/4 = (2s + 1)/4. Suppose y(δ(u)) ≤ 1/2, ∀u ∈ V (H). Then, \(y(E[T])\leq {\sum }_{u\in A(H)}y(\delta (u))\leq s/2\). Thus, y(E[T]) < (|E(T)| + 1)/4 in this case.

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Acknowledgements

The authors are grateful to the anonymous referees for a number of valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Toyohashi University of Technology, Toyohashi, 441-8580, Japan

    Toshihiro Fujito & Tomoaki Shimoda

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  1. Toshihiro Fujito
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  2. Tomoaki Shimoda
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Correspondence to Toshihiro Fujito.

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A preliminary version of this work appeared in Proc. 11th CSR (International Computer Science Symposium in Russia), 2016. This work is supported in part by JSPS KAKENHI under Grant Number 26330010.

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Fujito, T., Shimoda, T. On Approximating (Connected) 2-Edge Dominating Set by a Tree. Theory Comput Syst 62, 533–556 (2018). https://doi.org/10.1007/s00224-017-9764-y

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