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Fractional Linear Matroid Matching Is in Quasi-NC

Authors Rohit Gurjar, Taihei Oki , Roshan Raj

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Author Details

Rohit Gurjar
  • Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, India
Taihei Oki
  • Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Japan
Roshan Raj
  • Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, India

Funding

  • Gurjar, Rohit: funded by SERB MATRICS grant number MTR/2022/001009.
  • Oki, Taihei: funded by JSPS KAKENHI Grant Number JP22K17853 and JST ERATO Grant Number JPMJER1903.

Cite AsGet BibTex

Rohit Gurjar, Taihei Oki, and Roshan Raj. Fractional Linear Matroid Matching Is in Quasi-NC. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.63

Abstract

The matching and linear matroid intersection problems are solvable in quasi-NC, meaning that there exist deterministic algorithms that run in polylogarithmic time and use quasi-polynomially many parallel processors. However, such a parallel algorithm is unknown for linear matroid matching, which generalizes both of these problems. In this work, we propose a quasi-NC algorithm for fractional linear matroid matching, which is a relaxation of linear matroid matching and commonly generalizes fractional matching and linear matroid intersection. Our algorithm builds upon the connection of fractional matroid matching to non-commutative Edmonds' problem recently revealed by Oki and Soma (2023). As a corollary, we also solve black-box non-commutative Edmonds' problem with rank-two skew-symmetric coefficients.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parallel algorithms
  • Theory of computation → Algebraic complexity theory
Keywords
  • parallel algorithms
  • hitting set
  • non-commutative rank
  • Brascamp-Lieb polytope
  • algebraic algorithms

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