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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On Approximating DegreeBounded Network Design Problems
Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph G = (V, E) with edge costs c ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimumcost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasipolynomial time O(log²k/log log k)approximation algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general DegreeBounded Directed Steiner Tree (DBDST) problem, where we are additionally given a degree bound d_v on each vertex v ∈ V, and we require that every vertex v in the output tree has at most d_v children. We give a quasipolynomial time (O(log n log k), O(log² n))bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first nontrivial result for the problem. While our costguarantee is nearly optimal, the degree violation factor of O(log²n) is an O(log n)factor away from the approximation lower bound of Ω(log n) from the Set Cover hardness. The hardness result holds even on the special case of the DegreeBounded Group Steiner Tree problem on trees (DBGSTT). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an (O(log nlog k), O(log n))bicriteria approximation algorithm for DBGSTT.
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 Award ID(s):
 1910565
 NSFPAR ID:
 10300267
 Date Published:
 Journal Name:
 Leibniz international proceedings in informatics
 ISSN:
 18688969
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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