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Tight Bounds for Online Weighted Tree Augmentation

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Abstract

The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an n-vertex tree \(T = (V,E)\) and an additional set \(L \subseteq {V \atopwithdelims ()2}\) of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links F such that every terminal pair that has arrived so far is 2-edge-connected in \(T \cup F\). This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network design problem. They gave a deterministic \(O(\log ^2 n)\)-competitive algorithm and showed an \(\varOmega (\log n)\) lower bound on the competitive ratio of randomized algorithms. The case when T is a path is also interesting: it is exactly the online interval set cover problem, which also captures as a special case the parking permit problem studied by Meyerson (FOCS 2005). The contribution of this paper is to give tight results for online weighted tree and path augmentation problems. The main result of this work is a deterministic \(O(\log n)\)-competitive algorithm for online WTAP, which is tight up to constant factors.

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Notes

  1. In the online set cover problem, elements arrive online and need to be covered upon arrival by sets from a set system known in advance. (Note that not necessarily all elements will appear.)

  2. The interval set cover problem is a special case of the set cover problem where the elements are a set of points on the real line and the sets are intervals.

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Authors and Affiliations

  1. Technion, Haifa, Israel

    Joseph (Seffi) Naor

  2. The University of Sydney, Sydney, Australia

    Seeun William Umboh

  3. Cornell University, Ithaca, USA

    David P. Williamson

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  1. Joseph (Seffi) Naor
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  2. Seeun William Umboh
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  3. David P. Williamson
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Correspondence to Seeun William Umboh.

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This work was done in part while all authors were visiting the Simons Institute for the Theory of Computing, and while SWU was at Eindhoven University of Technology, the Hebrew University of Jerusalem and the Technion. JN was supported in part by ISF grant 1585/15 and BSF Grant 2014414. SWU was supported in part by NWO Grant 639.022.211 and ISF Grant 1817/17. A preliminary version of this paper appeared in ICALP 2019 [18]. This version includes proofs omitted from the preliminary version.

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Naor, J.(., Umboh, S.W. & Williamson, D.P. Tight Bounds for Online Weighted Tree Augmentation. Algorithmica 84, 304–324 (2022). https://doi.org/10.1007/s00453-021-00888-7

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  • DOI: https://doi.org/10.1007/s00453-021-00888-7