We study the problem of solving linear program in the streaming model. Given a constraint matrix A ∈ ℝ^{m×n} and vectors b ∈ ℝ^m, c ∈ ℝ^n, we develop a space-efficient interior point method that optimizes solely on the dual program. To this end, we obtain efficient algorithms for various different problems: - For general linear programs, we can solve them in Õ(√n log(1/ε)) passes and Õ(n²) space for an ε-approximate solution. To the best of our knowledge, this is the most efficient LP solver in streaming with no polynomial dependence on m for both space and passes. - For bipartite graphs, we can solve the minimum vertex cover and maximum weight matching problem in Õ(√m) passes and Õ(n) space. In addition to our space-efficient IPM, we also give algorithms for solving SDD systems and isolation lemma in Õ(n) spaces, which are the cornerstones for our graph results.
@InProceedings{liu_et_al:LIPIcs.ICALP.2023.88, author = {Liu, S. Cliff and Song, Zhao and Zhang, Hengjie and Zhang, Lichen and Zhou, Tianyi}, title = {{Space-Efficient Interior Point Method, with Applications to Linear Programming and Maximum Weight Bipartite Matching}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {88:1--88:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.88}, URN = {urn:nbn:de:0030-drops-181408}, doi = {10.4230/LIPIcs.ICALP.2023.88}, annote = {Keywords: Convex optimization, interior point method, streaming algorithm} }
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