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Efficiently Computing the Minimum Rank of a Matrix in a Monoid of Zero-One Matrices

Authors Stefan Kiefer , Andrew Ryzhikov

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Author Details

Stefan Kiefer
  • Department of Computer Science, University of Oxford, UK
Andrew Ryzhikov
  • Department of Computer Science, University of Oxford, UK

Funding

  • Ryzhikov, Andrew: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 852769, ARiAT).

Acknowledgements

We thank the anonymous reviewers for their helpful comments that improved the presentation of the paper.

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Stefan Kiefer and Andrew Ryzhikov. Efficiently Computing the Minimum Rank of a Matrix in a Monoid of Zero-One Matrices. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 61:1-61:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.61

Abstract

A zero-one matrix is a matrix with entries from {0, 1}. We study monoids containing only such matrices. A finite set of zero-one matrices generating such a monoid can be seen as the matrix representation of an unambiguous finite automaton, an important generalisation of deterministic finite automata which shares many of their good properties.
Let 𝒜 be a finite set of n×n zero-one matrices generating a monoid of zero-one matrices, and m be the cardinality of 𝒜. We study the computational complexity of computing the minimum rank of a matrix in the monoid generated by 𝒜. By using linear-algebraic techniques, we show that this problem is in NC and can be solved in 𝒪(mn⁴) time. We also provide a combinatorial algorithm finding a matrix of minimum rank in 𝒪(n^{2 + ω} + mn⁴) time, where 2 ≤ ω ≤ 2.4 is the matrix multiplication exponent. As a byproduct, we show a very weak version of a generalisation of the Černý conjecture: there always exists a straight line program of size 𝒪(n²) describing a product resulting in a matrix of minimum rank. 
For the special case corresponding to complete DFAs (that is, for the case where all matrices have exactly one 1 in each row), the minimum rank is the size of the smallest image of the set of states under the action of a word. Our combinatorial algorithm finds a matrix of minimum rank in time 𝒪(n³ + mn²) in this case.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Computing methodologies → Symbolic and algebraic manipulation
Keywords
  • matrix monoids
  • minimum rank
  • unambiguous automata

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