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Space-Efficient Interior Point Method, with Applications to Linear Programming and Maximum Weight Bipartite Matching

Authors S. Cliff Liu, Zhao Song, Hengjie Zhang, Lichen Zhang, Tianyi Zhou

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Author Details

S. Cliff Liu
  • Carnegie Mellon University, Pittsburgh, PA, USA
Zhao Song
  • Adobe Research, San Jose, CA, USA
Hengjie Zhang
  • Columbia University, New York, NY, USA
Lichen Zhang
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Tianyi Zhou
  • University of California San Diego, CA, USA

Funding

  • Zhang, Lichen: Supported by NSF grant No. CCF-1955217 and NSF grant No. CCF-2022448.

Acknowledgements

The authors would like to thank Jonathan Kelner for many helpful discussions.

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S. Cliff Liu, Zhao Song, Hengjie Zhang, Lichen Zhang, and Tianyi Zhou. Space-Efficient Interior Point Method, with Applications to Linear Programming and Maximum Weight Bipartite Matching. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 88:1-88:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.88

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Linear programming
Keywords
  • Convex optimization
  • interior point method
  • streaming algorithm

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