In this paper polynomiality of *n-fold integer programming problem* is discussed.

(My question is very simple and it is stated at the end. There is no need to deeply understand the excerpts they are provided for the purpose of being complete.)

Excerpt 1:

The *n-fold integer programming problem*. Fix a $ p \times q$ integer matrix $ A$ . Given positive integer $ n$ and integer vectors $ b = (b^0, b^1, \ldots , b^n)$ and $ c = (c^1, \ldots , c^n)$ , where $ b^0 \in \mathbb{Z}^q$ , and $ b^k \in \mathbb{Z}^p$ and $ c^k \in \mathbb{Z}^q$ for $ k = 1, \ldots, n$ , find a nonnegative integer vector $ x = (x^1, \ldots , x^n)$ , where $ x^k \in \mathbb{N}^q$ for $ k = 1, \ldots, n$ , which minimizes $ cx= \sum_{k=1}^{n}c^k x^k$ subject to the equations $ \sum_{k=1}^{n} x^k = b^0$ and $ Ax^k = b^k$ for $ k = 1, \ldots, n$ .

Excerpt 2:

In this article, we establish the following theorem. Naturally, the input size is $ n$ plus the bit size of the integer objective vector $ c \in \mathbb{Z}^{nq}$ and the integer right-hand side vector $ b \in \mathbb{Z}^{q+np}$ .

**Theorem 1.1.** *Fix any integer matrix $ A$ . Then there is a polynomial-time algorithm that, given any $ n$ and any integer vectors $ b$ and $ c$ , solves the corresponding n-fold integer programming problem.*

My Question:

I have an Integer Programming at hand. For a given integer $ k$ , my $ A$ (like above) is $ 3 \times 2k$ , and my $ n$ (like above) is $ k$ , and let us assume that bit size of my integers is of $ O(\log (k))$ . Is the aforementioned theorem still applicable to my situation? Especially because $ q$ in my situation depends on $ k.$