Derandomization for sparse approximations and independent sets

  • Thomas Hofmeister
  • Hanno Lefmann
Contributed Papers Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 969)


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/ε2) 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 Δ2−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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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Thomas Hofmeister
    • 1
  • Hanno Lefmann
    • 1
  1. 1.Lehrstuhl Informatik IIUniversität DortmundDortmundGermany

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