Column Subset Selection Problem is UG-hard

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 ? R m i? n and a positive integer k, and we want to select a sub-matrix C of k columns to minimize ? A - ? C A ? F , where ? C = C C + denotes the matrix of projection onto the space spanned by C. In the other problem, we are given A ? R m i? 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 ? A - C U R ? F , where U ? R c i? 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. 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.

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