# Derandomization for sparse approximations and independent sets

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## 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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## References

- [AEKS]M. Ajtai, P. Erdös, J. Komlós and E. Szemerédi,
*On Turán's Theorem for Sparse Graphs*, Combinatorica 1, 313–317 (1981).Google Scholar - [AKS1]M. Ajtai, J. Komlós and E. Szemerédi,
*A Note on Ramsey Numbers*, J. of Combinatorial Theory, Ser. A 29, 354–360 (1980).Google Scholar - [AKS2]M. Ajtai, J. Komlós and E. Szemerédi,
*A Dense Infinite Sidon Sequence*, European J. of Combinatorics 2, 2–11 (1981).Google Scholar - [A]I. Althöfer,
*On Sparse Approximations to Randomized Strategies and Convex Combinations*, Linear Algebra and its Applications 199, 339–355 (1994).Google Scholar - [AB]N. Alon and J. Bruck,
*Explicit Constructions of Depth*-2*Majority Circuits for Comparison and Addition*, SIAM J. on Discrete Mathematics 7, 1994, 1–8.Google Scholar - [ASE]N. Alon, J. Spencer and P. Erdös,
*The Probabilistic Method*, Wiley & Sons, New York (1992).Google Scholar - [BS]J. Bruck and R. Smolensky,
*Polynomial Threshold Functions, AC*^{0}*Functions and Spectral Norms*, SIAM J. on Computing 21, 33–42 (1992).Google Scholar - [HR]M. Halldórsson and J. Radhakrishnan,
*Greed is Good: Approximating Independent Sets in Sparse and Bounded-degree Graphs*, Proc. 26th ACM Symp. on Theory of Computing (STOC), 439–448 (1994).Google Scholar - [LY]R. J. Lipton and N. E. Young,
*Simple Strategies for Large Zero-Sum Games with Applications to Complexity Theory*, Proc. 26th ACM Symp. on Theory of Computing (STOC), 734–740 (1994).Google Scholar - [R]P. Raghavan,
*Probabilistic Construction of Deterministic Algorithms: Approximating Packing Integer Programs*, J. of Computer and System Sciences 37, 130–143 (1988).Google Scholar - [S1]J. Shearer,
*A Note on the Independence Number of Triangle-free Graphs*, Discrete Mathematics 46, 83–87 (1983).Google Scholar - [S2]J. Shearer,
*A Note on the Independence Number of Triangle-free Graphs, II*, J. of Combinatorial Theory 53, 300–307 (1991).Google Scholar - [Y]N. E. Young,
*Greedy Algorithms by Derandomizing Unknown Distributions*, preprint (1994). See also:*Randomized Rounding without Solving the Linear Program*, Proc. 6th ACM-SIAM Symp. on Discrete Algorithms (SODA), to appear (1995).Google Scholar