. A randomized strategy or a convex combination may be represented by a probability vector p = (p 1 ; : : : ; pm ) . p is called sparse if it has only few positive entries. This paper presents an Approximation Lemma and applies it to matrix games, linear programming, computer chess, and uniform sampling spaces. In all cases arbitrary probability vectors can be substituted by sparse ones (with only logarithmically many positive entries) without loosing too much performance. Short running title: Sparse Approximations 1 1. Introduction We present the following Approximation Lemma: Let A = (a ij ) be an m \Theta n--matrix over the real numbers with 0 a ij 1 for 1 i m , 1 j n . Let p = (p 1 ; : : : ; pm ) be a probability vector, i.e., 0 p i for all i and m X i=1 p i = 1 , and " ? 0 any positive constant. Then there exists another probability vector q = (q 1 ; : : : ; q m ) with at most k = l log 2n 2" 2 m many positive coordinates q i such that fi fi fi fi fi m X i=...

Approximation algorithms provide a natural way to approach computationally hard problems. There are currently many known paradigms in this area, including greedy algorithms, primal-dual methods, methods based on mathematical programming (linear and semidefinite programming in particular), local improvement, and "low distortion" embeddings of general metric spaces into special families of metric spaces. Randomization is a useful ingredient in many of these approaches, and particularly so in the form of randomized rounding of a suitable relaxation of a given problem. We survey this technique here, with a focus on correlation inequalities and their applications.

| Benjamin Doerr y Abstract This paper shows that the lattice approximation problem for totally unimodular matrices A 2 R mn can be solved eciently and optimally via a linear programming approach. The complexity of our algorithm is O(log m) times the complexity of nding an extremal point of a polytope in R n described by 2(m + n) linear constraints. We also consider the worst-case approximability called linear discrepancy. Here we derive an upper bound for the linear discrepancy of a totally unimodular m n matrix A: lindisc(A) minf1 1 n+1 ; 1 1 m g: This bound is sharp. It proves Spencer's conjecture lindisc(A) (1 1 n+1 ) herdisc(A) for totally unimodular matrices. It seems to be the rst time that linear programming is successfully used for a discrepancy problem. 1 Introduction and Results 1.1 Lattice Approximation Problem, Linear Discrepancy and Integer Linear Programs. Let A 2 R mn be any real matrix and b := Ap; p 2 R n , a point of the vector space ...

. In this article we introduce (combinatorial) multi--color discrepancy and generalize some classical results from 2--color discrepancy theory to c colors. We give a recursive method that constructs c--colorings from approximations to the 2--color discrepancy. This method works for a large class of theorems like the six--standard--deviation theorem of Spencer, the Beck--Fiala theorem and the results of Matousek, Welzl and Wernisch for bounded VC--dimension. On the other hand there are examples showing that discrepancy in c colors can not be bounded in terms of two--color discrepancy even if c is a power of 2. For the linear discrepancy version of the Beck--Fiala theorem the recursive approach also fails. Here we extend the method of floating colors to multi--colorings and prove multi--color versions of the the Beck--Fiala theorem and the Barany--Grunberg theorem. 1 Introduction Combinatorial discrepancy theory deals with the problem of partitioning the vertices of a hypergraph (set--...

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.

Abstract. Given n real numbers 0 ≤ x1,..., xn < 1 and a permutation σ of {1,..., n}, we can always find ¯x1,..., ¯xn ∈ {0, 1} so that the partial sums ¯x1 + · · · + ¯xk and ¯xσ1 + · · · + ¯xσk differ from the unrounded values x1 + · · · + xk and xσ1 + · · · + xσk by at most n/(n + 1), for 1 ≤ k ≤ n. The latter bound is best possible. The proof uses an elementary argument about flows in a certain network, and leads to a simple algorithm that finds an optimum way to round. Many combinatorial optimization problems in integers can be solved or approximately solved by first obtaining a real-valued solution and then rounding to integer values. Spencer [11] proved that it is always possible to do the rounding so that partial sums in two independent orderings are properly rounded. His proof was indirect—a corollary of more general results [7] about discrepancies of set systems—and it guaranteed only that the rounded partial sums would differ by at most 1 − 2−2n from the unrounded values. The purpose of this note is to give a more direct proof, which leads to a sharper result. Let x1,..., xn be real numbers and let σ be a permutation of {1,..., n}. We will write Sk = x1 + · · · + xk, Σk = xσ1 + · · · + xσk, 0 ≤ k ≤ n,

A (0, 1) matrix A is strongly unimodular if A is totally unimodular and every matrix obtained from A by setting a nonzero entry to 0 is also totally unimodular. Here we consider the linear discrepancy of strongly unimodular matrices. It was proved by Lovaz, et.al. [5] that for any matrix A, lindisc(A) # herdisc(A). (1) When A is the incidence matrix of a set-system, a stronger inequality holds: For any family H of subsets of {1, 2, . . . , n}, lindisc(H) # (1 - t n )herdisc(H). where t n # 2 -2 n (J. Spencer, [6]). In this paper we prove that the constant t n can be improved to 3 -(n+1)/2 for strongly unimodular matrices. # The first author is supported by NSF Grant DMS-9304580. + The second author is supported by Courant Instructorship, New York University. 1 1 Introduction and results A matrix A is said to be totally unimodular if the determinant of each square submatrix of A is 0 or 1. Clearly the entries of a totally unimodular matrix must be 0 or 1. A matr...

This survey paper focuses on my contributions to the area of polynomials with Littlewood-type coefficient constraints. It summarizes the main results from many of my recent papers some of which are joint with Peter Borwein.

We show that the linear discrepancy of a totally unimodular mn matrix A is at most lindisc(A) 1 1 n+1 : This bound is sharp. In particular, this result proves Spencer's conjecture lindisc(A) (1 1 n+1 ) herdisc(A) in the special case of totally unimodular matrices. If m 2, we also show lindisc(A) 1 1 m . Finally we give a characterization of those totally unimodular matrices which have linear discrepancy 1 1 n+1 : Besides m 1 matrices containing a single non-zero entry, they are exactly the ones which contain n + 1 rows such that each n thereof are linearly independent. A central proof idea is the use of linear programs. A preliminary version of this result appeared at SODA 2001. This work was partially supported by the graduate school `Eziente Algorithmen und Multiskalenmethoden', Deutsche Forschungsgemeinschaft y A similar result has been independently obtained by T. Bohman and R. Holzman and presented at the Conference on Hypergraphs (Gyula O.H. Katona is 60), Budapest, in June 2001. Mathematics Subject Classication (2000): Primary 11K38, 90C05. Secondary 05C65. Proposed abbreviated title: Linear Discrepancy. 2 1

We investigate a re ned recursive coloring approach to construct balanced colorings for hypergraphs. A coloring is called balanced if each hyperedge has (roughly) the same number of vertices in each color.