Improved approximation for spanning star forest in dense graphs

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 edges. We design a (α + (1−α)√c− 2)-approximation algorithm for spanning star forest in c-dense graphs for any 2 > 0, where α = 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 nodeand edgeweighted) 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 This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, and the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174. Part of this work was done while the authors were visiting Cornell University.