Let $$\langle a,x\rangle=b$$ be a linear constraint where $x\in\mathbb R^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\mathbb Z_{\geq0}$ (non-negative) and are of $m=poly(n)$ bit length.
Is there a polyhedron $By\leq c$ (depending on $a,b$) satisfying the properties
- $B\in\mathbb Z^{q\times(n+1)},c\in\mathbb Z^{q}$ where $q=poly(n)$
- $\log_2\max_{i,j}|B_{i,j}c_i|=O(n)$ ($O(n)$ bit length)
- $B,c$ constructible in $O(\log n)$ time in $poly(n)$ processors ($NC^1$ complexity)
- $\exists x\in\mathbb R_{\geq0}^n:\langle a,x\rangle=b\iff B[a,b]'\leq c$ or at least $B[a,b]'\leq c\implies \exists x\in\mathbb R_{\geq0}^n:\langle a,x\rangle=b$
satisfied?
I think there might be (perhaps every entry of $B,c$ can be computed in $O(\log n)$ time).
