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Sublinear-Round Parallel Matroid Intersection

Author Joakim Blikstad

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Author Details

Joakim Blikstad
  • KTH Royal Institute of Technology, Sweden

Funding

This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No 71567. The author is also supported by the Swedish Research Council (Reg. No. 2019-05622).

Acknowledgements

I thank Danupon Nanongkai and Sagnik Mukhopadhyay for insightful discussions and their valuable comments throughout the development of this work. Part of this work was done while the author visited BARC and the University of Copenhagen.

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Joakim Blikstad. Sublinear-Round Parallel Matroid Intersection. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.25

Abstract

Despite a lot of recent progress in obtaining faster sequential matroid intersection algorithms, the fastest parallel poly(n)-query algorithm was still the straightforward O(n)-round parallel implementation of Edmonds' augmenting paths algorithm from the 1960s.
Very recently, Chakrabarty-Chen-Khanna [FOCS'21] showed the lower bound that any, possibly randomized, parallel matroid intersection algorithm making poly(n) rank-queries requires Ω̃(n^{1/3}) rounds of adaptivity. They ask, as an open question, if the lower bound can be improved to Ω̃(n), or if there can be sublinear-round, poly(n)-query algorithms for matroid intersection.
We resolve this open problem by presenting the first sublinear-round parallel matroid intersection algorithms. Perhaps surprisingly, we do not only break the Õ(n)-barrier in the rank-oracle model, but also in the weaker independence-oracle model. Our rank-query algorithm guarantees O(n^{3/4}) rounds of adaptivity, while the independence-query algorithm uses O(n^{7/8}) rounds of adaptivity, both making a total of poly(n) queries.

Subject Classification

ACM Subject Classification
  • Theory of computation → Discrete optimization
  • Theory of computation → Parallel computing models
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Matroids and greedoids
Keywords
  • Matroid Intersection
  • Combinatorial Optimization
  • Parallel Algorithms

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References

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