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> Theory Seminar:Feb 6 *Cancelled*
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Theory Seminar:Feb 6 *Cancelled*
Event date: Friday, February 06, 2015, at 11:10 AM
Location: GB221
Speaker: Aleksandar Nikolov, Microsoft Research
Title: Randomized Rounding for the Largest j-Simplex Problem
Abstract:
The maximum volume j-simplex problem asks to compute the j-dimensional simplex of maximum volume inside the convex hull of a given set of npoints in d-dimensional space. This is a natural problem with long history in computational geometry, and falls in the class of problems of approximating an arbitrary body with a simpler one. We give adeterministic approximation algorithm for this problem which achievesan approximation ratio of e^{j/2 + o(j)} and runs in time polynomialin d and n. The problem is known to be NP-hard to approximate within afactor of c^j for some constant c. Our algorithm also approximates theproblem of finding the larges determinant principal j-by-j submatrixof a positive semidefinite matrix, with approximation ratio e^{j + o(j)}.This latter problem has connections with discrepancy theory and low-rankmatrix approximation. We achieve our approximation by rounding solutionsto a generalization of the D-optimal design problem, or, equivalently,the dual of an appropriate smallest enclosing ellipsoid problem.