A star graph is a tree of diameter at most two. A star forest is a graph that consists of node‐disjoint star graphs. In the spanning star forest problem, given an unweighted graph G, the objective is to find a star forest that contains all vertices of G and has the maximum number of edges. This problem is the complement of the dominating set problem in the following sense: On a graph with n vertices, the size of the maximum spanning star forest is equal to n minus the size of the minimum dominating set. We present a 0.71‐approximation algorithm for this problem, improving upon the approximation factor of 0.6 of Nguyen et al. (SIAM J. Comput. 38:946–962, 2008). We also present a 0.64‐approximation algorithm for the problem on node‐weighted graphs. Finally, we present improved hardness of approximation results for the weighted (both edge‐weighted and node‐weighted) versions of the problem. Our algorithms use a non‐linear rounding scheme, which might be of independent interest.
