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DOI: 10.4230/LIPIcs.STACS.2026.16

To Buy or Not to Buy: Online Rent-Or-Buy on Node-Weighted Graphs

Authors: Sander Borst and Moritz Venzin

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

We study the rent-or-buy variant of the online Steiner forest problem on node- and edge-weighted graphs. For n-node graphs with at most ̄{n} nodes of non-zero weight, and at most k̃ different arriving terminal pairs, we obtain the following: - A deterministic, O(log n log ̄{n})-competitive algorithm against adaptive adversaries. This improves on the previous best, O(log⁴ n)-competitive algorithm obtained by the black-box reduction from [Yair Bartal et al., 2001] combined with the previously best deterministic algorithms for the simpler "buy-only" setting. - A deterministic, O(̄{n}log k̃)-competitive algorithm against adaptive adversaries. This generalizes the O(log k̃)-competitive algorithm for the purely edge-weighted setting from [Seeun Umboh, 2015]. - A randomized, O(log k̃ log ̄{n})-competitive algorithm against oblivious adversaries. All previous approaches were based on the randomized, black-box reduction from [Awerbuch et al., 2004] that achieves a O(log k̃ log n)-competitive ratio when combined with an algorithm for the "buy-only" setting. Our key technical ingredient is a novel charging scheme to an instance of online prize-collecting set cover. This allows us to extend the witness-technique of [Seeun Umboh, 2015] to the node-weighted setting and obtain refined guarantees with respect to ̄{n}, already in the much simpler "buy-only" setting.

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Sander Borst and Moritz Venzin. To Buy or Not to Buy: Online Rent-Or-Buy on Node-Weighted Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{borst_et_al:LIPIcs.STACS.2026.16,
  author =	{Borst, Sander and Venzin, Moritz},
  title =	{{To Buy or Not to Buy: Online Rent-Or-Buy on Node-Weighted Graphs}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{16:1--16:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.16},
  URN =		{urn:nbn:de:0030-drops-255054},
  doi =		{10.4230/LIPIcs.STACS.2026.16},
  annote =	{Keywords: online network design, node-weighted Steiner forest, derandomization}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2021.73

Majorizing Measures for the Optimizer

Authors: Sander Borst, Daniel Dadush, Neil Olver, and Makrand Sinha

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract

The theory of majorizing measures, extensively developed by Fernique, Talagrand and many others, provides one of the most general frameworks for controlling the behavior of stochastic processes. In particular, it can be applied to derive quantitative bounds on the expected suprema and the degree of continuity of sample paths for many processes. One of the crowning achievements of the theory is Talagrand’s tight alternative characterization of the suprema of Gaussian processes in terms of majorizing measures. The proof of this theorem was difficult, and thus considerable effort was put into the task of developing both shorter and easier to understand proofs. A major reason for this difficulty was considered to be theory of majorizing measures itself, which had the reputation of being opaque and mysterious. As a consequence, most recent treatments of the theory (including by Talagrand himself) have eschewed the use of majorizing measures in favor of a purely combinatorial approach (the generic chaining) where objects based on sequences of partitions provide roughly matching upper and lower bounds on the desired expected supremum. In this paper, we return to majorizing measures as a primary object of study, and give a viewpoint that we think is natural and clarifying from an optimization perspective. As our main contribution, we give an algorithmic proof of the majorizing measures theorem based on two parts: - We make the simple (but apparently new) observation that finding the best majorizing measure can be cast as a convex program. This also allows for efficiently computing the measure using off-the-shelf methods from convex optimization. - We obtain tree-based upper and lower bound certificates by rounding, in a series of steps, the primal and dual solutions to this convex program. While duality has conceptually been part of the theory since its beginnings, as far as we are aware no explicit link to convex optimization has been previously made.

Cite as

Sander Borst, Daniel Dadush, Neil Olver, and Makrand Sinha. Majorizing Measures for the Optimizer. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 73:1-73:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{borst_et_al:LIPIcs.ITCS.2021.73,
  author =	{Borst, Sander and Dadush, Daniel and Olver, Neil and Sinha, Makrand},
  title =	{{Majorizing Measures for the Optimizer}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{73:1--73:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.73},
  URN =		{urn:nbn:de:0030-drops-136120},
  doi =		{10.4230/LIPIcs.ITCS.2021.73},
  annote =	{Keywords: Majorizing measures, Generic chaining, Gaussian processes, Convex optimization, Dimensionality Reduction}
}

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To Buy or Not to Buy: Online Rent-Or-Buy on Node-Weighted Graphs

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Majorizing Measures for the Optimizer

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