This question is an instance of the well-known closest point problem for lattices, but I am having difficulty interpreting the standard complexity results in this context.
We are given a generator matrix for an $m$-dimensional sublattice of ${\mathbb Z}^n$ with $m < n$. But I want to think of this data as constant - so in particular $m$ and $n$ are constants, and you can do as much preprocessing as you like on it.
The input to the problem is a vector $v= (v_1,\ldots,v_n) \in {\mathbb Z}^n$, and I want to find a closest point to $v$ in the lattice using the $L_1$-norm. That is, a lattice point $(x_1,\ldots,x_n)$ that minimizes $\sum_{i=1}^n|v_i-x_i|$.
My question is, can this be done in time polynomial in the size of $v$, that is polynomial in $\sum_{i=1}^n \log |v_i| : 1 \le i \le n $?