eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-09-01
20:1
20:14
10.4230/LIPIcs.ESA.2022.20
article
Resource Sharing Revisited: Local Weak Duality and Optimal Convergence
Blankenburg, Daniel
1
https://orcid.org/0000-0002-2469-3206
Research Institute for Discrete Mathematics, UniversitΓ€t Bonn, Germany
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 π_β-minimal element in a Minkowski sum π³ = β_{C β π} X_C of non-empty closed convex sets X_C β β^β_{β₯ 0}, where π and β 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 π_{β}-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 Ξ©((π|+|β|)/Ξ΄Β² log |β|) 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol244-esa2022/LIPIcs.ESA.2022.20/LIPIcs.ESA.2022.20.pdf
Resource sharing
Dantzig-Wolfe-type algorithms
Decreasing minimization