Tree Embedding and Directed Steiner Problems

Tree-Embedding is one of the most powerful techniques in the area of approximation algorithms as it reduces many diffcult problems like group Steiner tree (GST) on general graphs into amenable tree intances. However, the developments on tree-embedding are pertained only to undirected graphs, thus rendering it useless against problems on directed graphs like directed Steiner tree (DST) and its generalization k-edge connected directed Steiner tree (k-DST). The latter problem, k-DST, is a notorious problem that has no known non-trivial approximation algorithm despites having been mentioned many times in literature, e.g., by Feldman et al. [SODA’09, JCSS’12], by Cheriyan et al. [SODA’12, TALG’14] and by Laekhanukit [SODA’14]. We explore the possibility of obtaining a non-trivial approximation algorithm for k-DST via tree-embedding. Let us briefly define k-DST. Given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k, the goal is to find a min-cost subgraph H ⊆ G that connects r to each terminal t ∈ T by k edge-disjoint r, t-paths. We reduce k-GST on general graphs into instances on trees. As a result, we devise an O(D·k ·log h)-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on “general graphs” where an instance has a D-shallow optimal solution, i.e., there exists k edge-disjoint r, t-paths, each of length at most D, for every terminal t ∈ T . Our contribution is two fold. First, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We remark that the k-hardness instance of k-DST is a DAG with 5 layers, and our algorithm gives O(k log h)-approximation for this special case. Second, we initiate the study of tree-embedding on directed graphs, which might later have more applications.