Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Directed Network Design Problems
针对有向网络设计问题，提出拟多项式时间的近似算法，在预算约束下最大化单调子模函数，改进了有向斯坦纳树等问题的近似比或运行时间。
We consider the following general network design problem. The input is an asymmetric metric (V, c), root [Formula: see text], monotone submodular function [Formula: see text], 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 [Formula: see text]-approximation algorithm for this problem, in which [Formula: see text] is the number of vertices in an optimal solution. As a consequence, we obtain an [Formula: see text]-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 [Formula: see text]-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 but improves significantly on the running 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 nontrivial approximation ratio. Under certain complexity assumptions, our approximation ratios are the best possible (up to constant factors).