Column subset selection is NP-complete

Let M be a real r×c matrix, and let k be a positive integer. In the column subset selection problem (CSSP), we need to minimize the quantity ‖M−SA‖, where A can be an arbitrary k×c matrix, and S runs over all r×k submatrices of M. This problem and its applications in numerical linear algebra are being discussed for several decades, but its algorithmic complexity remained an open issue. We show that CSSP is NP-complete.