Perfect out-forest problem and directed Steiner cycle packing problem

Abstract

The perfect out-forest problem generalizes the perfect matching problem, and the directed Steiner cycle packing problem generalizes the Hamiltonian cycle decomposition problem and is a variant of the directed Steiner tree packing problem. In this paper, we study the complexity of these two types of digraph packing problems.

For the perfect out-forest problem, Gutin and Yeo proved that it is NP-complete to decide whether a given strong digraph contains a 0-perfect out-forest. We show that this result also holds for the 1-perfect out-forest problem. However, when restricted to a semicomplete digraph $D$, the problem of deciding whether $D$ contains an $i$-perfect out-forest becomes polynomial-time solvable, where $i\in \{0,1\}$. In addition, we prove that it is NP-hard to find a 0-perfect out-forest of maximum size in a connected acyclic digraph, and it is NP-hard to find a 1-perfect out-forest of maximum size in a connected digraph.

For the directed Steiner cycle packing problem, when both $k\ge 2,\ell \ge 1$ are fixed integers, we show that the problem of deciding whether there are at least $\ell$ internally disjoint directed $S$-Steiner cycles in a digraph $D$ is NP-complete, where $S\subseteq V\left(D\right)$ and $\left|S\right|=k$. However, when we consider the class of symmetric digraphs, the problem becomes polynomial-time solvable. We also show that the problem of deciding whether there are at least $\ell$ arc-disjoint directed $S$-Steiner cycles in a given digraph $D$ is NP-complete, where $S\subseteq V\left(D\right)$ and $\left|S\right|=k$.