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

Rohan Ghuge, Viswanath Nagarajan

Abstract: We consider the following general network design problem on directed graphs. The input is an asymmetric metric (V, c), root r* ϵ V, monotone submodular function f: 2V → ℝ+ and budget B. The goal is to find an r*-rooted arborescence T of cost at most B that maximizes f (T). Our main result is a very simple quasi-polynomial time -approximation algorithm for this problem, where k ≤ |V| 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 -approximation algorithm for directed (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 -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 time whereas the previous algorithm required 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).