How to select a subset of columns in a given matrix to satisfy a condition?
The feasibility version of your problem is NP-complete, for any fixed $t &gt; 1$, by reduction from SAT. 

Suppose we are given a formula over the variables $x_1,\ldots,x_a$, having clauses $C_1,\ldots,C_b$. We will have $2a$ columns, one column per literal. 

For each variable $x_i$ we will have the following row:


Positions $x_i,\lnot x_i$ have value $1$, and all other columns have the value $1/t$.


This row forces the solution $S$ not to be a singleton, and forces it to contain exactly one of $\{x_i,\lnot x_i\}$.

For each clause $C_j = \ell_1 \lor \cdots \lor \ell_c$ we will have the following row:


Positions $\ell_1,\ldots,\ell_c$ have values $1,1/t,\ldots,1/t^{c-1}$, and all other positions have value $1/t^c$.


This row forces the solution to contain at least one of the literals $\ell_1,\ldots,\ell_c$.

Conversely, any satisfying assignment corresponds to a set $S$, as can be easily checked. Hence a set $S$ exists iff the original formula is satisfiable.