Quick Approximation to Matrices and Applications

  title={Quick Approximation to Matrices and Applications},
  author={Alan M. Frieze and Ravi Kannan},
m×n matrix A with entries between say −1 and 1, and an error parameter ε between 0 and 1, we find a matrix D (implicitly) which is the sum of simple rank 1 matrices so that the sum of entries of any submatrix (among the ) of (A−D) is at most εmn in absolute value. Our algorithm takes time dependent only on ε and the allowed probability of failure (not on m, n).We draw on two lines of research to develop the algorithms: one is built around the fundamental Regularity Lemma of Szemerédi in Graph… 

On Non-Approximability for Quadratic Programs Preliminary Version

This paper studies the computational complexity of the following type of quadratic programs and shows that it is quasi-NP-hard to approximate to a factor better than O(log n) for some γ > 0.

N ov 2 01 9 Matrix Decompositions and Sparse Graph Regularity ∗

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It is proved that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix with a polylogarithmic number of blocks, achieving better error bound than the existing decomposition theorem of Frieze and Kannan.

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  • A. FriezeR. KannanS. Vempala
  • Computer Science
    Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
  • 1998
This paper develops an algorithm which is qualitatively faster provided the entries of the matrix are sampled according to a natural probability distribution and the algorithm takes time polynomial in k, 1//spl epsiv/, log(1//spl delta/) only, independent of m, n.

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  • N. AlonA. FriezeD. Welsh
  • Mathematics, Computer Science
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
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  • A. FriezeR. Kannan
  • Mathematics
    Proceedings of 37th Conference on Foundations of Computer Science
  • 1996
The central point here is that the Regularity Lemma provides an explanation of why these Max-SNP hard problems turn out to be easy in dense graphs.

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