Minimize the maximum component of a sum of vectors
We cannot discuss the complexity of a problem when the problem size is fixed to a constant, because (most part of) the complexity theory deals with the asymptotic behavior of the complexity of the problem as the problem size tends to infinity.  Here, I consider both the number of rows and the dimention of the vectors as variables.

Then the problem is NP-complete even if the numbers in the input are given in unary.  This is not an answer to your question because you are asking about approximation, but it is something.

Define the problem rigorously:

Instance: n pairs of vectors ai, bi ∈ ℕm (i ∈ {1, …, n}), and K ∈ ℕ, all in unary.
Question: Can we choose either ai or bi for each i so that the sum of these n vectors has at most K in each coordinate?

The following is a known NP-complete problem called 3-partition:

3-partition
Instance: B ∈ ℕ, and 3k integers c1, …, c3k between B/4 and B/2, exclusive, such that ∑i=13k ci = kB, all in unary.
Question: Can the multiset {c1, …, c3k} be partitioned into k multisets S1, …, Sk such that the sum of each Sj is equal to B?

Given an instance (B; c1, …, c3k) of the 3-partition problem, construct an instance of the above problem as follows.  For each i = 1, …, 3k and j = 1, …, k, we will construct a pair of 4k-dimensional vectors, representing the choice on whether ci belongs to Sj or not:


The vector representing the choice “ci ∈ Sj” has only one nonzero entry, which is (k−1)ci at the jth coordinate.
The vector representing the choice “ci ∉ Sj” also has only one nonzero entry, which is B at the (k+i)-th coordinate.


It is not hard to see that the instance (B; c1, …, c3k) of the 3-partition problem has a solution if and only if there is a way to choose a vector from each of the 3k2 constructed pairs so that all the coordinates of the sum of these vectors are at most (k−1)B.  (In fact, when this happens, all the coordinates of the sum are equal to (k−1)B.)  So this is a reduction from the 3-partition problem to the above problem.

So far, I ignored the two additional constraints stated at the end of the question, but both are easy to enforce by modifying this reduction slightly.  The condition that the sum of the elements of each vector is equal can be enforced by appending dummy coordinates which contain only 0 or 1.  The condition that this sum is less than the dimension can be enforced by appending dummy coordinates which contain only 0.