Packing paths in digraphs

Let be a fixed set of digraphs. Given a digraph H, a -packing in H is a collection of vertex disjoint subgraphs of H, each isomorphic to a member of . A -packing is maximum if the number of vertices belonging to members of is maximum, over all -packings. The analogous problem for undirected graphs has been extensively studied in the literature. The purpose of this paper is to initiate the study of digraph packing problems. We focus on the case when is a family of directed paths. We show that unless is (essentially) either , or , the G-packing problem is NP-complete. When , the -packing problem is simply the matching problem. We treat in detail the one remaining case, . We give in this case a polynomial algorithm for the packing problem. We also give the following positive results: a Berge type augmenting configuration theorem, a min-max characterization, and a reduction to bipartite matching. These results apply also to packings by the family consisting of all directed paths and cycles. We also explore weighted variants of the problem and include a polyhedral analysis

Packing Paths in Digraphs

Let G be a fixed collection of digraphs. Given a digraph H, a G- packing of H is a collection of vertex disjoint subgraphs of H, each isomorphic to a member of G. A G-packing, P, is maximum if the number of vertices belonging to some member of P is maximum, over all G-packings. The analogous problem for undirected graphs has been extensively studied in the literature. We concentrate on the cases when G is a family of paths. We show G-packing is NP-complete when (essentially) G is not one of the families f ~ P 1 g, or f ~ P 1 ; ~ P 2 g: When G = f ~ P 1 g, the G-packing problem is simply the matching problem. The main focus of our paper is the case when G = f ~ P 1 ; ~ P 2 g, the directed paths of length one and two. We present a collection of augmenting configurations such that a packing is maximum if and only if it contains no augmenting configuration. We also present a min-max condition which yields a concise certificate for maximality of packings. We apply these results to obtain..

Sidepath results on packing P⃗1\vec{P}_1's and P⃗2\vec{P}_2's

This report provides proofs and refinements of some results from our companion paper [1: Packing paths in digraphs, R.C. Brewster, P. Hell, S.H. Pantel, R. Rizzi, A. Yeo, Journal of Graph Theory 2003]. We refer the reader to [1] for a more detailed introduction to notation, background, and motivation. Let $\cal G$ be a fixed set of digraphs. Given a digraph $H$, a $\cal G$-packing in $H$ is a collection $\cal P$ of vertex disjoint subgraphs of $H$, not necessarily induced, each isomorphic to a member of $\cal G$. A $\cal G$-packing $\cal P$ is {\em maximum} if the number of vertices belonging to members of $\cal P$ is maximum, over all $\cal G$-packings. The analogous problem for undirected graphs has been extensively studied in the literature. In a companion paper we initiate the study of digraph packing problems, focusing on the case when $\cal G$ is a family of directed paths. We showed that unless $\cal G$ is (essentially) either $\{ \vec{P}_1 \}$, or $\{ \vec{P}_1, \vec{P}_2 \}$, the $\cal G$-packing problem is NP-complete. We use the notation $\vec{P}_k$ for the \emph{directed path of length $k$}, i.e., the path $u_0, u_1, \dots, u_k$ in which all arcs are oriented from $u_{i-1}$ to $u_i$ for $i=1, 2, \dots k$. When ${\cal G} = \{ \vec{P}_1 \}$, the $\cal G$-packing problem is simply the matching problem. In [2: On the complexity of digraph packings, R.C. Brewster, R. Rizzi, IPL 2003], we treat in detail the one remaining case, ${\cal G} = \{ \vec{P}_1, \vec{P}_2 \}$. We give in this case a polynomial time algorithm based on augmenting configurations, and a corresponding Berge-type and Tutte-type theorems. We also give a reduction to bipartite matching. In this report, we give a direct combinatorial algorithm based on augmentations, explore weighted variants of the problem, and give a polyhedral analysis