Annals of Combinatorics

, Volume 19, Issue 4, pp 661–669 | Cite as

Markov Bases and Generalized Lawrence Liftings

  • Hara Charalambous
  • Apostolos Thoma
  • Marius Vladoiu


Minimal Markov bases of configurations of integer vectors correspond to minimal binomial generating sets of the assocciated lattice ideal. We give necessary and sufficient conditions for the elements of a minimal Markov basis to be (a) inside the universal Gröbner basis and (b) inside the Graver basis. We study properties of Markov bases of generalized Lawrence liftings for arbitrary matrices \({A \in \mathcal{M}_{m \times n}(\mathbb{Z})}\) and \({B \in \mathcal{M}_{p \times n}(\mathbb{Z})}\) and show that in cases of interest the complexity of any two Markov bases is the same.

Mathematics Subject Classification

14M25 14L32 13P10 62H17 


toric ideals Markov basis Graver basis Lawrence liftings 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Hara Charalambous
    • 1
  • Apostolos Thoma
    • 2
  • Marius Vladoiu
    • 3
  1. 1.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece
  2. 2.Department of MathematicsUniversity of IoanninaIoanninaGreece
  3. 3.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharestRomania

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