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# Approximation Algorithms for Directed Weighted Spanners

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## Acknowledgements

We thank several anonymous reviewers for their valuable comments and suggestions that improved the quality of the writeup.

## Cite As

Elena Grigorescu, Nithish Kumar, and Young-San Lin. Approximation Algorithms for Directed Weighted Spanners. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 8:1-8:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.8

## Abstract

In the pairwise weighted spanner problem, the input consists of a weighted directed graph on n vertices, where each edge is assigned both a cost and a length. Furthermore, we are given k terminal vertex pairs and a distance constraint for each pair. The goal is to find a minimum-cost subgraph in which the distance constraints are satisfied. We study the weighted spanner problem, in which the edges have positive integral lengths of magnitudes that are polynomial in n, while the costs are arbitrary non-negative rational numbers. Our results include the following in the classical offline setting: - An Õ(n^{4/5 + ε})-approximation algorithm for the weighted pairwise spanner problem. When the edges have unit costs and lengths, the best previous algorithm gives an Õ(n^{3/5 + ε})-approximation, due to Chlamtáč, Dinitz, Kortsarz, and Laekhanukit (Transactions on Algorithms, 2020). - An Õ(n^{1/2+ε})-approximation algorithm for the weighted spanner problem when the terminal pairs consist of all vertex pairs and the distances must be preserved exactly. When the edges have unit costs and arbitrary positive lengths, the best previous algorithm gives an Õ(n^{1/2})-approximation for the all-pair spanner problem, due to Berman, Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and Computation, 2013). We also prove the first results for the weighted spanners in the online setting. Our results include the following: - An Õ(k^{1/2 + ε})-competitive algorithm for the online weighted pairwise spanner problem. The state-of-the-art results are an Õ(n^{4/5})-competitive algorithm when edges have unit costs and arbitrary positive lengths, and a min{Õ(k^{1/2 + ε}), Õ(n^{2/3 + ε})}-competitive algorithm when edges have unit costs and lengths, due to Grigorescu, Lin, and Quanrud (APPROX, 2021). - An Õ(k^ε)-competitive algorithm for the online weighted single-source (or single-sink) spanner problem. Without distance constraints, this problem is equivalent to the online directed Steiner tree problem. The best previous algorithm for online directed Steiner trees is an Õ(k^ε)-competitive algorithm, due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018). Our online results also imply efficient approximation algorithms for the corresponding offline problems. To the best of our knowledge, these are the first approximation (online) polynomial-time algorithms with sublinear approximation (competitive) ratios for the weighted spanner problems.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Online algorithms
• Theory of computation → Routing and network design problems
• Theory of computation → Rounding techniques
##### Keywords
• directed weighted spanners
• linear programming
• junction tree

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