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Resource Sharing Revisited: Local Weak Duality and Optimal Convergence

Authors: Daniel Blankenburg

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)


Abstract
We revisit the (block-angular) min-max resource sharing problem, which is a well-known generalization of fractional packing and the maximum concurrent flow problem. It consists of finding an 𝓁_∞-minimal element in a Minkowski sum 𝒳 = ∑_{C ∈ 𝒞} X_C of non-empty closed convex sets X_C ⊆ ℝ^ℛ_{≥ 0}, where 𝒞 and ℛ are finite sets. We assume that an oracle for approximate linear minimization over X_C is given. We improve on the currently fastest known FPTAS in various ways. A major novelty of our analysis is the concept of local weak duality, which illustrates that the algorithm optimizes (close to) independent parts of the instance separately. Interestingly, this implies that the computed solution is not only approximately 𝓁_{∞}-minimal, but among such solutions, also its second-highest entry is approximately minimal. Based on a result by Klein and Young [Klein and Young, 2015], we provide a lower bound of Ω((𝒞|+|ℛ|)/δ² log |ℛ|) required oracle calls for a natural class of algorithms. Our FPTAS is optimal within this class - its running time matches the lower bound precisely, and thus improves on the previously best-known running time for the primal as well as the dual problem.

Cite as

Daniel Blankenburg. Resource Sharing Revisited: Local Weak Duality and Optimal Convergence. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{blankenburg:LIPIcs.ESA.2022.20,
  author =	{Blankenburg, Daniel},
  title =	{{Resource Sharing Revisited: Local Weak Duality and Optimal Convergence}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.20},
  URN =		{urn:nbn:de:0030-drops-169582},
  doi =		{10.4230/LIPIcs.ESA.2022.20},
  annote =	{Keywords: Resource sharing, Dantzig-Wolfe-type algorithms, Decreasing minimization}
}
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