Adaptive Sparsification for Matroid Intersection

Author Kent Quanrud



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Kent Quanrud
  • Dept. of Computer Science, Purdue University, West Lafayette, IN, USA

Acknowledgements

We thank the reviewers for their careful and helpful feedback. We thank Adrian Vladu for teaching us about [Assadi, 2024].

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Kent Quanrud. Adaptive Sparsification for Matroid Intersection. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 118:1-118:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.118

Abstract

We consider the matroid intersection problem in the independence oracle model. Given two matroids over n common elements such that the intersection has rank k, our main technique reduces approximate matroid intersection to logarithmically many primal-dual instances over subsets of size Õ(k). This technique is inspired by recent work by [Assadi, 2024] and requires additional insight into structuring and efficiently approximating the dual LP. This combination of ideas leads to faster approximate maximum cardinality and maximum weight matroid intersection algorithms in the independence oracle model. We obtain the first nearly linear time/query approximation schemes for the regime where k ≤ n^{2/3}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Discrete optimization
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Matroid intersection
  • adaptive sparsification
  • multiplicative-weight udpates
  • primal-dual

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