2,011 research outputs found

    Approximating Hereditary Discrepancy via Small Width Ellipsoids

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    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 this bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time O~(log3n)\tilde{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 simple O(log3/2n)O(\log^{3/2} n)-approximation algorithm for this problem. We show that up to this approximation factor, the hereditary discrepancy of a matrix AA is characterized by the optimal value of simple geometric convex program that seeks to minimize the largest \ell_{\infty} norm of any point in a ellipsoid containing the columns of AA. This characterization promises to be a useful tool in discrepancy theory

    Antimatroids and Balanced Pairs

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    We generalize the 1/3-2/3 conjecture from partially ordered sets to antimatroids: we conjecture that any antimatroid has a pair of elements x,y such that x has probability between 1/3 and 2/3 of appearing earlier than y in a uniformly random basic word of the antimatroid. We prove the conjecture for antimatroids of convex dimension two (the antimatroid-theoretic analogue of partial orders of width two), for antimatroids of height two, for antimatroids with an independent element, and for the perfect elimination antimatroids and node search antimatroids of several classes of graphs. A computer search shows that the conjecture is true for all antimatroids with at most six elements.Comment: 16 pages, 5 figure

    On largest volume simplices and sub-determinants

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    We show that the problem of finding the simplex of largest volume in the convex hull of nn points in Qd\mathbb{Q}^d can be approximated with a factor of O(logd)d/2O(\log d)^{d/2} in polynomial time. This improves upon the previously best known approximation guarantee of d(d1)/2d^{(d-1)/2} by Khachiyan. On the other hand, we show that there exists a constant c>1c>1 such that this problem cannot be approximated with a factor of cdc^d, unless P=NPP=NP. % This improves over the 1.091.09 inapproximability that was previously known. Our hardness result holds even if n=O(d)n = O(d), in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where cˉ\bar c is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a d×nd\times n matrix

    Some relational structures with polynomial growth and their associated algebras II: Finite generation

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    The profile of a relational structure RR is the function φR\varphi_R which counts for every integer nn the number, possibly infinite, φR(n)\varphi_R(n) of substructures of RR induced on the nn-element subsets, isomorphic substructures being identified. If φR\varphi_R takes only finite values, this is the Hilbert function of a graded algebra associated with RR, the age algebra A(R)A(R), introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure RR admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra.Comment: 27 pages; submitte
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