Improved approximation for spanning star forest in dense graphs

  • Authors:

  • Jing He;Hongyu Liang

  • Affiliations:

  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China FIT 4-609;Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China FIT 4-609

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2013

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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驴(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 驴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驴(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.