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A fast algorithm to construct a representation for transversal matroids

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Abstract

Transversal matroids were introduced in the mid 1960s and unified many results in transversal theory. Piff and Welsh proved that a transversal matroid is representable over all sufficiently large fields. To date, their merge algorithm is the only known algorithm to construct a representation matrix for a given transversal matroid. In this paper, a new algorithm to construct a representation matrix for a given transversal matroid is proposed that is faster than the Piff–Welsh algorithm. Let \(G=(V^{(1)}\dot{\cup }V^{(2)},E)\) be the bipartite graph representing a transversal matroid and \({\mathcal {M}}\) the set of all complete matchings of G. The time complexity of the proposed algorithm is \(O\left( |V^{(1)}|^{1/2}|E| + |V^{(1)}| |{\mathcal {M}}|\right) \). This algorithm makes use of complete matchings of bipartite graphs instead of matrix determinants, and an enumeration algorithm is used to find these matchings.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

    Morteza Rekab-Eslami & Morteza Esmaeili

  2. Department of Electrical and Computer Engineering, University of Victoria, PO Box 1700, STN CSC, Victoria, BC, V8W 2Y2, Canada

    Morteza Esmaeili & T. Aaron Gulliver

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  1. Morteza Rekab-Eslami
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  2. Morteza Esmaeili
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  3. T. Aaron Gulliver
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Correspondence to T. Aaron Gulliver.

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Rekab-Eslami, M., Esmaeili, M. & Gulliver, T.A. A fast algorithm to construct a representation for transversal matroids. Japan J. Indust. Appl. Math. 33, 207–226 (2016). https://doi.org/10.1007/s13160-016-0209-9

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  • DOI: https://doi.org/10.1007/s13160-016-0209-9

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