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].