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2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS) (2015)
Berkeley, CA, USA
Oct. 17, 2015 to Oct. 20, 2015
ISSN: 0272-5428
ISBN: 978-1-4673-8191-8
pp: 230-249
In this paper, we consider the following inverse maintenance problem: given A ε R{n×d} and a number of rounds r, at round k, we receive a n×n diagonal matrix D(k) and we wish to maintain an efficient linear system solver for AT D(k) A under the assumption D(k) does not change too rapidly. This inverse maintenance problem is the computational bottleneck in solving multiple optimization problems. We show how to solve this problem with Õ(nnz(A)+d&#x03A9) preprocessing time and amortized Õ(nnz(A)+d2) time per round, improving upon previous running times. Consequently, we obtain the fastest known running times for solving multiple problems including, linear programming, computing a rounding of a polytope, and sampling a point in a polytope. In particular given a feasible point in a linear program with n variables, d constraints, and constraint matrix A ε R{d×n}, we show how to solve the linear program in time Õ((nnz(A)+d2)√d log(ε -- 1)). We achieve our results through a novel combination of classic numerical techniques of low rank update, preconditioning, and fast matrix multiplication as well as recent work on subspace embeddings and spectral sparsification that we hope will be of independent interest.
Maintenance engineering, Linear systems, Linear programming, Stability analysis, Approximation algorithms, Approximation methods, Optimization

Y. T. Lee and A. Sidford, "Efficient Inverse Maintenance and Faster Algorithms for Linear Programming," 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), Berkeley, CA, USA, 2015, pp. 230-249.
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