Binary matrix column subset selection complexity
Rephrasing as a set system, each row represents a subset $E_i$ of some set $X$, for $i=1,2,\dots,m$.  You want a set $Y \subseteq X$ with at most $k$ elements, such that $E_i \cap Y \ne \emptyset$ for each $i$.  In other words, you want a hitting set of size at most $k$; this problem is NP-complete.