How to partition the rows of a matrix in such a way that every column satisfies a given condition?
As suggested by Yuval Filmus, reduce PARTITION to my problem. 

Given an instance of PARTITION, that is a set of nonnegative integers $\{b_1, \ldots, b_k\}$, is there a subset $S\subset\{1,\ldots,k\}$, such that $\sum_{i\in S}b_i=\sum_{i\notin S}b_i=\frac{\sum_{i=1}^kb_i}{2}$?

Let $n=2$, $m=k$, $a_{ij}=b_i$ for all $(i,j)\in\{1,\ldots,k\}\times\{1,2\}$ and $V_1=V_2=\frac{\sum_{i=1}^kb_i}{2}$.

This is clearly created in polynomial-time. 

PARTITION is solved if and only if my problem is solved. 


If PARTITION is solved: there is a set $S\subset\{1,\ldots,k\}$, such that $\sum_{i\in S}b_i=\sum_{i\notin S}b_i=\frac{\sum_{i=1}^kb_i}{2}$. Take $S_1=S$ and $S_2=\{1,\ldots,k\}\backslash S$. Clearly, $S_1\cup S_2=\{1,\ldots,k\}$ and $S_1$ and $S_2$ are disjoint. Further, we have
$$\sum_{i\in S_1}a_{i1}=\sum_{i\in S_1}b_i=V_1\geqslant V_1,\\
\sum_{i\in S_2}a_{i2}=\sum_{i\in S_2}b_i=V_2\geqslant V_2,$$
and my problem is solved. 
If my problem is solved: there are disjoint $S_1$ and $S_2$ such that 
$$S_1\cup S_2=\{1,\ldots,k\},\\
\sum_{i\in S_1}a_{i1}=\sum_{i\in S_1}b_i\geqslant V_1,\\
\sum_{i\in S_2}a_{i2}=\sum_{i\in S_2}b_i\geqslant V_2.$$
Since $V_1=V_2=\frac{\sum_{i=1}^kb_i}{2}$ and $\sum_{i\in S_1}b_i+\sum_{i\in S_2}b_i=\sum_{i=1}^kb_i$, we must have 
$$\sum_{i\in S_1}b_i=\sum_{i\notin S_2}b_i=\frac{\sum_{i=1}^kb_i}{2},$$
and PARTITION is solved.


Therefore, my problem is NP-hard. 

To solve the problem, let us write it as integer programming problem as suggested by Yuval Filmus. To do so, introduce the binary variable $x_{ij}$ that is equal to $1$, if $i$ is in set $S_j$, and, $0$ otherwise.

\begin{align}
      &amp; {\underset{\mathbf{ x }}{\text{maximize}}}
      &amp; &amp; 0\\[6pt]
      &amp; \text{subject to}
      &amp; &amp; \sum_{i=1}^ma_{ij}x_{ij}\geqslant V_j,\forall\, j\in\{1,\ldots,n\},\tag{C1}\\[6pt]
      &amp; &amp; &amp; \sum_{j=1}^nx_{ij}=1, \forall\, i\in\{1,\ldots,m\},\tag{C2}\\[6pt]
      &amp; &amp; &amp; x_{ ij }\in\{0, 1\}, \forall (i,j)\in\{1,\ldots,k\}\times\{1,\ldots,n\}\tag{C3}.
    \end{align}

Even though this solves my problem, I need to develop a greedy algorithm for it, can I do that?