Approximating Directed Buy-at-Bulk Network Design

Antonakopoulos, Spyridon

doi:10.1007/978-3-642-18318-8_2

Cited by 4 publications

(16 citation statements)

References 22 publications

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“…As a direct application of Theorem 1.2, we obtain an O( log 2 k log log k )-approximation for single-source buy-at-bulk network design. This improves over the previous best O(log 3 k)-approximation [Ant11]. Buy-at-bulk network design is a well-studied generalization of Steiner tree that involves concave cost-functions on edges.…”

Section: Find An Approximately Optimalmentioning

confidence: 87%

“…The non-uniform problem is also hard to approximate better than O(log log n) [CGNS08]. For the directed case that we consider, the only prior result is [Ant11] which implies a quasi-polynomial time O(log 3 k)-approximation for the non-uniform version. Buy-at-bulk problems have also been studied for multi-commodity flows [CHKS10], which we do not consider in this paper.…”

Section: Related Workmentioning

confidence: 96%

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

Ghuge

Nagarajan

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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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 : 2 V → 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 log log 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( log 2 k log log 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( log 2 k log log 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 log 1+ǫ 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.

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Section: Find An Approximately Optimalmentioning

confidence: 87%

Section: Related Workmentioning

confidence: 96%

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

Ghuge

Nagarajan

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…While most problems admit polylogarithmic approximations in either the online or offline setting for undirected networks [11,21,33,54], the problems are much harder for directed networks. In the offline setting, the current best approximation ratio is O(k ε ) for the directed Steiner tree problem [30,76], O(min{k 1/2+ε , n 2/3+ε }) for the directed Steiner forest problem [20,33], and O(min{k 1/2+ε , n 4/5+ε }) for the directed buy-at-bulk problem [6]. In the online setting for directed networks, [29] showed that compared to offline, it suffices to pay an extra polylogarithmic factor, where the polylogarithmic term was later improved by [73].…”

Section: Additional Background and Related Workmentioning

confidence: 99%

“…Namely, x is good if there is at least one (fractional) s i t i path of length at most d i . In LP (6), the feasibility problem is equivalent to the following problem. Given the local graph G i and edge weight z, is there an s i t i path of length at most d i and weight less than 1?…”

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confidence: 99%

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Online Directed Spanners and Steiner Forests

Grigorescu¹,

Lin²,

Quanrud³

2021

Preprint

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We present online algorithms for directed spanners and directed Steiner forests. These are well-studied network connectivity problems that fall under the unifying framework of online covering and packing linear programming formulations. This framework was developed in the seminal work of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and is based on primal-dual techniques. Specifically, our results include the following:

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