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Breaking O(nr) for Matroid Intersection

Author Joakim Blikstad

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Subject Classification

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

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Abstract

We present algorithms that break the Õ(nr)-independence-query bound for the Matroid Intersection problem for the full range of r; where n is the size of the ground set and r ≤ n is the size of the largest common independent set. The Õ(nr) bound was due to the efficient implementations [CLSSW FOCS'19; Nguyên 2019] of the classic algorithm of Cunningham [SICOMP'86]. It was recently broken for large r (r = ω(√n)), first by the Õ(n^{1.5}/ε^{1.5})-query (1-ε)-approximation algorithm of CLSSW [FOCS'19], and subsequently by the Õ(n^{6/5}r^{3/5})-query exact algorithm of BvdBMN [STOC'21]. No algorithm - even an approximation one - was known to break the Õ(nr) bound for the full range of r. We present an Õ(n√r/ε)-query (1-ε)-approximation algorithm and an Õ(nr^{3/4})-query exact algorithm. Our algorithms improve the Õ(nr) bound and also the bounds by CLSSW and BvdBMN for the full range of r.

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Joakim Blikstad. Breaking O(nr) for Matroid Intersection. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.31

Author Details

Joakim Blikstad
  • KTH Royal Institute of Technology, Stockholm, 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.

Acknowledgements

I thank Danupon Nanongkai and Sagnik Mukhopadhyay for insightful discussions and their valuable comments throughout the development of this work.

References

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