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 the 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.
[9]. 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 versions of the problem.