# Derandomization for sparse approximations and independent sets

## Abstract

It is known (see Althöfer [A]) that for every n×m-matrix *A* with entries taken from the interval [0, 1] and for every probability vector *p*, there is a sparse probability vector *q* with only *O*(ln *n/ε*^{2}) nonzero entries such that every component of the vector *A*·*q* differs from every component of *A* · *p* in absolute value by at most *ε*. In [A], the existence of such a vector is proved by a probabilistic argument. It is stated as an open problem whether there is an efficient, i.e. polynomial-time, deterministic algorithm which actually constructs such a vector *q*.

In this paper, we provide such an algorithm which takes time polynomial in *n,m*, and 1/*ε*. The algorithm is based on the method of “pessimistic estimators”, introduced by Raghavan [R].

Moreover, we apply a similar derandomization strategy to the Independent Set Problem for graphs with not too many triangles. Improving recent results of Halldórsson and Radhakrishnan [HR], we give an efficient algorithm which computes an independent set of size \(\Omega (\frac{n}{\Delta }ln \Delta )\)for a graph *G* on *n* vertices with maximum degree *Δ*, if *G* contains only a little less than the maximum possible number of triangles (say *n Δ*^{2−ε} many for a positive constant *ε*). This algorithm is based on earlier results concerning the independence number of triangle-free graphs due to Ajtai, Komlós, Szemerédi [AKS1] and Shearer [S1].

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