Column Subset Selection Problem is UG-hard

Highlights

• Select a subset of columns/rows of a matrix so that they represent the matrix well.

• Formulated as Column Subset Selection Problem and Column–Row Subset Selection Problem.

• Unique Games Conjecture implies that there is no PTAS.

• First complexity theoretic result of this kind for these problems.

Abstract

We address two problems related to selecting an optimal subset of columns from a matrix. In one of these problems, we are given a matrix $A\in {\mathbb{R}}^{m\times n}$ and a positive integer k, and we want to select a sub-matrix C of k columns to minimize ${\Vert A-{\Pi }_{C}A\Vert }_F$, where ${\Pi }_C=C{C}^{+}$ denotes the matrix of projection onto the space spanned by C. In the other problem, we are given $A\in {\mathbb{R}}^{m\times n}$, positive integers c and r, and we want to select sub-matrices C and R of c columns and r rows of A, respectively, to minimize ${\Vert A-CUR\Vert }_F$, where $U\in {\mathbb{R}}^{c\times r}$ is the pseudo-inverse of the intersection between C and R. Although there is a plethora of algorithmic results, the complexity of these problems has not been investigated thus far. We show that these two problems are NP-hard assuming UGC.