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Approximate Hypergraph Coloring

Kameli, Hamid | 2011

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 42564 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahmoodian, Ebadollah; Hajiabolhassan, Hossein
  7. Abstract:
  8. Problem of 2-colorability of hypergraphs was introduced by Erd?os in 1963, and there are many results about this problem until now. In this thesis we consider this problem from the algorithmic viewpoint. In all of the algorithms we assume that the given hypergraph is 2-colorable and we want to color it. By using a simple construction, it was shown that for every r = 3, the problem of coloring of hypergraphs is as hard as the problem of coloring graphs. If NP ?= ZPP12, it is impossible to approximate the chromatic number of r-uniform hyhpergraphs with n vertices by a factor of n1-? for any fixed ? > 0, in time polynomial of n. In reference [KS03] a coloring algorithm with approximation factor of O(n(log log n)2 (log n)2 ), for r-uniform hypergraphs is introduced. Even, in the case that hypergraph is 3- uniform,Dinur, Regev and Smyth prove that for any constant c, coloring a hypergraph with c colors is NP-Hard. In reference [Alo96], For the special case of hypergraphs of dimension three, an approximation algorithm with O(n 2 9 log 17 8 n) colors is introduced. In reference [Che96] For a class of dense 3-uniform 2-colorable hypergraphs, introduce a randomized coloring algorithm with 2 color, too. Then for a model of random 2-colorable hypergraphs, by applying a spectral method, a randomized algorithm is introduced, which can almost surely find a proper 2-coloring
  9. Keywords:
  10. Approximate Algorithm ; Hypergraph Coloring ; Randomized Rounding ; Random Hypergraph ; Semidefinite Programming

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