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Table 1 Comparison of the number of variables and equations and of the dimension of the resulting solution space (nullspace) of the FLB and NB dual system and their corresponding MILP

From: Speeding up the core algorithm for the dual calculation of minimal cut sets in large metabolic networks

  FLB dual system [Eq. (7)] Generalized NB dual system [Eq. (13)]
# dual variables \(m + n + \left| {Irrev} \right| + t\) \(n + t\)
# (in)equalities \(n + 1\) \(n - m + 1 \enspace \enspace\)(*)
Dimension of solution space (nullspace) of the dual system \(m + \left| {Irrev} \right| + t\) \(m + t\)
  FLB MILP [Eq. (14)] NB MILP [Eq. (16)]
# variables in the corresponding MILP Continuous: \(2n + m + t\) Continuous: \(2n + t\)
  Binary: \(n\) Binary: \(n\)
# (in)equalities in the corresponding MILP \(n + 1 + m + d\) \(\left( {n - m} \right) + 1 + m + d = n + 1 + d\)
  1. The number of variables of the MILP includes the variables for the desired system integrated in the MILP, whereas the number of (in)equalities of the MILP excludes (flux) bounds and indicator constraints. \(m\): number of metabolites; \(n\): number of reactions; \(t\): number of rows (inequalities) in matrix \({\mathbf{T}}\)/vector \({\mathbf{t}}\) in Eq. (5);\({ }d\): number of rows (inequalities) in matrix \({\mathbf{D}}\)/vector \({\mathbf{d}};\) \(\left| {Irrev} \right|\) number of irreversible reactions. (*) It is assumed that the stoichiometric matrix \({\mathbf{N}}\) has full row rank (conservation relations removed), i.e. rank(\({\mathbf{N}}\)) = \({ }m\).