Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Other Directed Network Design Problems

We consider the following general network design problem on directed graphs. The input is an asymmetric metric (V, c), root r* is an element of V, monotone submodular function f : 2(V) -> R+ and budget B. The goal is to find an r*-rooted arborescence T of cost at most B that maximizes integral(T). Our main result is a very simple quasi-polynomial time O(log k/log log k) approximation algorithm for this problem, where k <= vertical bar V vertical bar is the number of vertices in an optimal solution. To the best of our knowledge, this is the first non-trivial approximation ratio for this problem. As a consequence we obtain an O(log(2) k/log log k) approximation algorithm for directed log log k (polymatroid) Steiner tree in quasi-polynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved O(log(2) k/log log k)-approximation algorithms for the single-source buy-at-bulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio [15], but improves significantly on the running time: our algorithm takes n(O)(log(1)(+epsilon k)) time whereas the previous algorithm required n(O)(log(5) k) time. For polymatroid Steiner tree and single-source buy-at-bulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first non-trivial approximation ratio. Under certain complexity assumptions, our approximation ratios are best possible (up to constant factors).