2026-07-24T17:52:31Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16958 2024-03-06T10:58:49Z ddc:004 open_access

Resource Sharing Revisited: Local Weak Duality and Optimal Convergence Blankenburg, Daniel Resource sharing Dantzig-Wolfe-type algorithms Decreasing minimization 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Daniel Blankenburg 2022 Is Part Of LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2022.20 urn:nbn:de:0030-drops-169582 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.20 eng https://creativecommons.org/licenses/by/4.0/legalcode