In other words, you want to find a permutation $\sigma$ that minimizes $|\sum_i c_{i\sigma(i)}|$?
That is NP-hard, by reduction from the subset sum problem, which asks, for a finite (multi)set of numbers $a_1,\ldots,a_n$, whether there is a nonempty subset that sums to 0.
Given an instance of subset sum, construct a $c_{ij}$ matrix of dimension $(2n-1)\times(2n-1)$ as follows: $C=\begin{bmatrix} a_1 & a_1 & \cdots & a_1 & 0 & \cdots & 0 \\ a_2 & a_2 & \cdots & a_2 & 0 & \cdots & 0 \\ \vdots & \vdots &\ddots& \vdots & \vdots && \vdots \\ a_n & a_n & \cdots & a_n & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 & 0 & \cdots & 0 \\ \vdots & \vdots && \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 0 & \cdots & 0 \end{bmatrix}$ with $n$ copies of each $a_i$, $n-1$ columns of zeroes to the right, and $n-1$ rows of zeroes at the bottom.
Then $\min_\sigma |\sum_i c_{i\sigma(i)}|=0$ if and only if there's a nonempty subset of the $a_i$s that sums to $0$. (Clearly $\sum_i c_{i\sigma(i)}$ is always the sum of some subset of the $a_i$s, and it has to be a nonempty subset because there are not enough zero columns for the permutation to avoid all of them).