Improved approximation for spanning star forest in dense graphs

Jing He () and Hongyu Liang ()
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Jing He: Tsinghua University
Hongyu Liang: Tsinghua University

Journal of Combinatorial Optimization, 2013, vol. 25, issue 2, No 6, 255-264

Abstract: Abstract A spanning subgraph of a graph G is called a spanning star forest of G if it is a collection of node-disjoint trees of depth at most 1. The size of a spanning star forest is the number of leaves in all its components. The goal of the spanning star forest problem is to find the maximum-size spanning star forest of a given graph. In this paper, we study the spanning star forest problem on c-dense graphs, where for any fixed c∈(0,1), a graph of n vertices is called c-dense if it contains at least cn 2/2 edges. We design a $(\alpha+(1-\alpha)\sqrt{c}-\epsilon)$ -approximation algorithm for spanning star forest in c-dense graphs for any ϵ>0, where $\alpha=\frac{193}{240}$ is the best known approximation ratio of the spanning star forest problem in general graphs. Thus, our approximation ratio outperforms the best known bound for this problem when dealing with c-dense graphs. We also prove that, for any constant c∈(0,1), approximating spanning star forest in c-dense graphs is APX-hard. We then demonstrate that for weighted versions (both node- and edge-weighted) of this problem, we cannot get any approximation algorithm with strictly better performance guarantee on c-dense graphs than on general graphs. Finally, we give strong inapproximability results for a closely related problem, namely the minimum dominating set problem, restricted on c-dense graphs.

Keywords: Spanning star forest; Dense graph; Approximation algorithm; Hardness of approximation (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10878-012-9499-2

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