We consider the following general network design problem on directed graphs. The input is an asymmetric metric (V, c), root r in V, monotone submodular function f and budget B. The goal is to find an rrooted arborescence T of cost at most B that maximizes f(T). Our main result is a very simple quasipolynomial 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 nontrivial approximation ratio for this problem. As a consequence we obtain an O(log^2 k / loglog k) approximation algorithm for directed (polymatroid) Steiner tree in quasipolynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved approximation algorithms for the singlesource buyatbulk 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 singlesource buyatbulk, 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 best possible (up to constant factors).
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QuasiPolynomial Algorithms for Submodular Tree Orienteering and Directed Network Design Problems
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 rrooted arborescence T of cost at most B that maximizes f(T). Our main result is a simple quasipolynomial 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 quasipolynomial 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 singlesource buyatbulk 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 singlesource buyatbulk, 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).
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 NSFPAR ID:
 10383440
 Date Published:
 Journal Name:
 Mathematics of Operations Research
 Volume:
 47
 Issue:
 2
 ISSN:
 0364765X
 Page Range / eLocation ID:
 1612 to 1630
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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