The Discrepancy of a hypergraph is the minimum attainable value, over twocolorings of its vertices, of the maximum absolute imbalance of any hyperedge. The Hereditary Discrepancy of a hypergraph, de\ufb01ned as the maximum discrepancy of a restriction of the hypergraph to a subset of its vertices, is a measure of its complexity. Lova\u00b4sz, 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. Matou\u02c7sek (2011) showed that this bound is tight up to a polylogarithmic factor, leaving open the question of actually computing this bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time \u02dc O(log3 n)-approximation to hereditary discrepancy, as a by-product of their work in di\ufb00erential privacy. In this paper, we give a direct simple O(log3\/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 \u2113\u221e norm of any point in a ellipsoid containing the columns of A. This characterization promises to be a useful tool in discrepancy theory.<\/p>\n","protected":false},"excerpt":{"rendered":"
The Discrepancy of a hypergraph is the minimum attainable value, over twocolorings of its vertices, of the maximum absolute imbalance of any hyperedge. The Hereditary Discrepancy of a hypergraph, de\ufb01ned as the maximum discrepancy of a restriction of the hypergraph to a subset of its vertices, is a measure of its complexity. Lova\u00b4sz, Spencer and 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