Abstract

. 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=...

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