### DIMACS Theoretical Computer Science Seminar

Title: Approximating Hereditary Discrepancy via Small Width Ellipsoids

Speaker: **Aleksandar Nikolov**, Rutgers University

Date: Wednesday, January 29, 2014 11:00-12:00pm

Location: CoRE Bldg, Room 301A, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

The Discrepancy of a hypergraph is the minimum attainable value, over two-colorings of its vertices, of the maximum absolute imbalance of any hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum discrepancy of a restriction of the hypergraph to a subset of its vertices, is a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related the natural extension of this quantity to matrices to rounding algorithms for linear programs, and gave a determinant based lower bound on the hereditary discrepancy. Matousek (2011) showed that this bound is tight up to a polylogarithmic factor, leaving open the question of actually computing the bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time O(log^3 n)-approximation to hereditary discrepancy, as a by-product of their work in differential privacy. In this paper, we give a direct and simple O(log^{3/2} n)-approximation algorithm for this problem. We show that up to this approximation factor, the hereditary discrepancy of a matrix A is characterized by the optimal value of simple geometric convex program that seeks to minimize the largest infinity norm of any point in a ellipsoid containing the columns of A.

Joint work with Kunal Talwar.

See: http://www.math.rutgers.edu/~sk1233/theory-seminar/S14/