Tl;dr: what is a max support flux vector sequence?
In a paper (arXiv link; full citation below) on continuous chemical reaction networks (CCRNs)$^1$ it discusses an algorithm for reaching different states using various chemical reactions. I have some questions about a line of the algorithm and what it means given the authors' notation.
Preliminaries
The paper defines a set of species $\Lambda$ (which is finite) and a set of reactions over $\Lambda$ as $R$, which is also finite. A reaction is an element $\rho =(\mathbf{r},\mathbf{p})\in\mathbb{N}^\Lambda\times\mathbb{N}^\Lambda$ ($\mathbb{N}$ being the set of all natural numbers); $\rho$ as a reaction seems to be $\in R$ (which makes sense) as I understand the paper. Here $\mathbf{r}$ and $\mathbf{p}$ represent the stoichometry of the reactants and products. A state of a CCRN is defined as a vector $\mathbf{c}\in\mathbb{R}^\Lambda_{\geq 0}$ specifying a non-negative concentration of each species.
A flux vector is also defined, as a vector $\mathbf{u}\in\mathbb{R}^R_{\geq 0}$ (the paper also defines a support of a flux vector and whether or not a flux vector is applicable at a state $\mathbf{c}$). Finally, a flux vector sequence $\mathbf{U}$ is defined as a tuple of flux vectors.
Algorithm
The algorithm takes the input of $\Lambda$ (and by consequence $R$) as well as the starting state of the system $\mathbf{c}$ and the desired ending state of the system $\mathbf{d}$. The relevant portions of the algorithm (not exactly quoted) are as follows (the numbers at the left signify the line numbers as listed in the paper):
2: for each reaction $\rho\in R$
3: if a vector $F_\rho\in\mathbb{Q}^R_{\geq 0}$ such that certain conditions are true compute it
4: if no such vector exists, eliminate $\rho$ from $R$
5: end for
8: for each $\rho\in R$ set $S(\rho) = \frac{1}{|R|}\sum^\limits{|R|}_\limits{i=1}F_i(\rho)$
9: compute $\epsilon = \frac{\text{min}\{S(\rho)\}_{\rho\in R}}{2}$
10: compute the max support flux vector sequence $\mathbf{U}_{\mathbf{c},\epsilon}$
Question(s)
Nowhere in the paper does it say anything about a max support flux vector sequence - what does it mean by this, and how would one calculate it? I am also unsure what a flux vector or a flux vector sequence represents physically speaking - I understand most of the other concepts defined, but not these at all. Another related question can be found at How do subscripts affect this calculation? (I was advised to split the two questions up).
$^1$Case, Adam; Lutz, Jack H.; Stull, D.M., Reachability problems for continuous chemical reaction networks, ZBL06630580.