Let $A$ be a $n \times n$ matrix with non-negative entries $a_{ij}$, where $a_{ij}$ is the entry in the $i^{th}$ row and $j^{th}$ column. Assume $\sum_{1 \leq j \leq n} a_{ij} \leq 1$ for all $1 \leq i \leq n$. Also assume $a_{ii} = 0$ for all $1 \leq i \leq n$.

I want to partition the index set $I = \{1, 2 \ldots n\}$ into minimum number of sets $I_1, I_2, \ldots I_t$ so that the column sum is bounded by $1$ in each sub-matrix defined by the sets, or more formally:

- $\cup_{1 \leq k \leq t} I_k = I$
- For all $1 \leq k \leq t$, $\sum_{i \in I_k}a_{ij} \leq 1$ for all $j \in I_k$
- The number $t$ is minimized

I can construct examples where $t$ has to be at least $2$, on the other hand, $t = \Theta(\log n)$ would suffice for all such matrices. I am wondering if a tighter bound exists.

Motivation: this is a sort of generalization of the coloring problem in bounded out-degree digraphs. If a di-graph has out-degree upper bounded by $k$ it can be colored with $k + 1$ colors.