On Variants of the Spanning Star Forest Problem

Abstract

A star forest is a collection of vertex-disjoint trees of depth at most 1, and its size is the number of leaves in all its components. A spanning star forest of a given graph G is a spanning subgraph of G that is also a star forest. The spanning star forest problem (SSF for short) [14] is to find the maximum-size spanning star forest of a given graph. In this paper, we study several variants of SSF, obtaining first or improved approximation and hardness results in all settings as shown below.

• 1 We study SSF on graphs of minimum degree δ(n), n being the number of vertices in the input graph. Call this problem ( ≥ δ(n))-SSF. We give an (almost) complete characterization of the complexity of ( ≥ δ(n))-SSF with respect to δ(n) as follows.

  • When 1 ≤ δ(n) ≤ O(1), ( ≥ δ(n))-SSF is APX-complete.

  • When ω(1) ≤ δ(n) ≤ O(n ^1 − ε) for some constant ε > 0, ( ≥ δ(n))-SSF is NP-hard but admits a PTAS.

  • When δ(n) ≥ ω(n ^1 − ε) for every constant ε > 0, ( ≥ δ(n))-SSF is not NP-hard assuming Exponential Time Hypothesis (ETH).

• 2 We investigate the spanning k-tree forest problem, which is a natural generalization of SSF. We obtain the first inapproximability bound of $1-\Omega(\frac{1}{k})$ for this problem, which asymptotically matches the known approximation ratio of $1-\frac{1}{k+1}$ given in [13]. We then propose an approximation algorithm for it with a slightly improved approximation ratio of $1-\frac{1}{k+2}$.

• 3 We prove that SSF cannot be approximated to any factor larger than $\frac{244}{245}$ in polynomial time, unless P = NP. This improves the previously best known bound of $\frac{259}{260}$ [14].

This work was supported in part by the National Basic Research Program of China Grant 2007CB807900, 2007CB807901, the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174. Part of this work was done while the authors were visiting Cornell University.