Online Directed Spanners and Steiner Forests
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:
- For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{4/5}) for graphs with general edge lengths, where n is the number of vertices of the given graph. For graphs with uniform edge lengths, we give an efficient randomized algorithm with competitive ratio Õ(n^{2/3+ε}), and an efficient deterministic algorithm with competitive ratio Õ(k^{1/2+ε}), where k is the number of terminal pairs. To the best of our knowledge, these are the first online algorithms for directed spanners. In the offline version, the current best approximation ratio for uniform edge lengths is Õ(n^{3/5 + ε}), due to Chlamt{á}č, Dinitz, Kortsarz, and Laekhanukit (SODA 2017, TALG 2020).
- For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized algorithm with competitive ratio Õ(n^{2/3 + ε}). The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is Õ(k^{1/2 + ε})-competitive. In the offline version, the current best approximation ratio with uniform costs is Õ(n^{26/45 + ε}), due to Abboud and Bodwin (SODA 2018).
To obtain efficient and competitive online algorithms, we observe that a small modification of the online covering and packing framework by Buchbinder and Naor implies a polynomial-time implementation of the primal-dual approach with separation oracles, which a priori might perform exponentially many calls to the oracle. We convert the online spanner problem into an online covering problem and complete the rounding-step analysis in a problem-specific fashion.
online directed pairwise spanners
online directed Steiner forests
online covering/packing linear programming
primal-dual approach
Theory of computation~Online algorithms
Theory of computation~Packing and covering problems
Theory of computation~Routing and network design problems
Theory of computation~Rounding techniques
5:1-5:25
APPROX
E.G and Y.L. were supported in part by NSF CCF-1910659 and NSF CCF-1910411.
https://arxiv.org/abs/2103.04543
We thank the anonymous reviewers for comments and suggestions that helped improve the presentation. We thank Anupam Gupta and Greg Bodwin for bringing to our attention references that we missed in previous versions of the writeup.
Elena
Grigorescu
Elena Grigorescu
Department of Computer Science, Purdue University, West Lafayette, IN, USA
https://www.cs.purdue.edu/homes/egrigore/
https://orcid.org/0000-0001-9673-4313
Young-San
Lin
Young-San Lin
Department of Computer Science, Purdue University, West Lafayette, IN, USA
https://www.cs.purdue.edu/homes/lin532/
https://orcid.org/0000-0002-5719-6708
Kent
Quanrud
Kent Quanrud
Department of Computer Science, Purdue University, West Lafayette, IN, USA
https://kentquanrud.com/
10.4230/LIPIcs.APPROX/RANDOM.2021.5
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Elena Grigorescu, Young-San Lin, and Kent Quanrud
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