2018/12/05 by Ghuge, Rohan, Nagarajan, Viswanath
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.1812.01768
We consider the following general network design problem on directed graphs. The input is an asymmetric metric (V,c), root r*∈ 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 simple quasi-polynomial time O((log k)/(loglog k))-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 O((log2 k)/(loglog k))-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 O((log2 k)/(loglog 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 [GLL19]. Our algorithm has the advantage of being deterministic and faster: the runtime is exp(O(log n log1+ε k)). 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. All our approximation ratios are tight (up to constant factors) for quasi-polynomial algorithms.