## On selecting a maximum volume sub-matrix of a matrix and related problems.(English)Zbl 1181.15002

Given a real $$m\times n$$ matrix, the authors consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible. The notion is important in low-rank approximations of matrices and rank revealing QR factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways.
In the present paper, from a complexity theoretic point of view, the authors propose four related problems. They establish the NP-hardness of these problems and they further show that they do not admit PTAS.

### MSC:

 15A03 Vector spaces, linear dependence, rank, lineability 15A12 Conditioning of matrices
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### References:

 [1] Businger, P.A.; Golub, G.H., Linear least squares solutions by Householder transformations, Numerische Mathematik, 7, 269-276, (1965) · Zbl 0142.11503 [2] Chandrasekaran, S.; Ipsen, I.C.F., On rank-revealing factorizations, SIAM journal of matrix analysis and its applications, 15, 592-622, (1994) · Zbl 0796.65030 [3] Chan, T.F., Rank revealing QR factorizations, Linear algebra appl., 88/89, 67-82, (1987) · Zbl 0624.65025 [4] Deshpande, A.; Rademacher, L.; Vempala, S.; Wang, G., Matrix approximation and projective clustering via volume sampling, (), 1117-1126 · Zbl 1192.68889 [5] Deshpande, A.; Vempala, S., Adaptive sampling and fast low-rank matrix approximation, (), 292-303 · Zbl 1155.68575 [6] de Hoog, F.R.; Mattheijb, R.M.M., Subset selection for matrices, Linear algebra and its applications, 422, 349-359, (2007) · Zbl 1158.15017 [7] Drineas, P.; Kannan, R.; Mahoney, M.W., Fast Monte Carlo algorithms for matrices III: computing a compressed approximate matrix decomposition, SIAM journal on computing, 36, 1, 184-206, (2006) · Zbl 1111.68149 [8] Drineas, P.; Kannan, R.; Mahoney, M.W., Fast Monte Carlo algorithms for matrices II: computing a low-rank approximation to a matrix, SIAM journal on computing, 36, 1, 158-183, (2006) · Zbl 1111.68148 [9] Frieze, A.; Kannan, R.; Vempala, S., Fast Monte-Carlo algorithms for finding low-rank approximations, Journal of the association for computing machinery, 51, 6, 1025-1041, (2004) · Zbl 1125.65005 [10] Garey, M.R.; Johnson, D.S., Computers and intractability, (1979), W. H. Freeman · Zbl 0411.68039 [11] G.H. Golub, V. Klema, G.W. Stewart, Rank degeneracy and least squares problems, Dept. of Computer Science, Univ. of Maryland, 1976 [12] Golub, G.H.; Loan, C.V., Matrix computations, (1996), Johns Hopkins U. Press [13] Goreinov, S.A.; Tyrtyshnikov, E.E., (), 47-51 [14] Gu, M.; Eisenstat, S.C., Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM journal on scientific computing, 17, 4, 848-869, (1996) · Zbl 0858.65044 [15] Hong, Y.P.; Pan, C.T., Rank-revealing QR factorizations and the singular value decomposition, Mathematics of computation, 58, 213-232, (1992) · Zbl 0743.65037 [16] Kahan, W., Numerical linear algebra, vol. 9, (1966), pp. 757-801 · Zbl 0236.65025 [17] Karp, R.M., Reducibility among combinatorial problems, (), 85-103 · Zbl 0366.68041 [18] Lenstra, A.K.; Lenstra, H.W.; Lovasz, L., Factoring polynomials with rational coefficients, Mathematische annalen, 261, 515-534, (1982) · Zbl 0488.12001 [19] Pan, C.T.; Tang, P.T.P., Bounds on singular values revealed by QR factorizations, BIT numerical mathematics, 39, 740-756, (1999) · Zbl 0944.65042 [20] Pan, C.T., On the existence and computation of rank-revealing $$L U$$ factorizations, Linear algebra and its applications, 316, 1-3, 199-222, (2000) · Zbl 0962.65023
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