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Approximation Algorithms for the Matroid-Constraint Maximum Subdeterminant

Kermani, Fatemeh | 2019

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 52570 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Ebrahimi Brojeni, Javad
  7. Abstract:
  8. let Mn n be a positive semmi definite matrix and M:= (U; B) be a matroid. Consider the problem of finding S 2 B such that det(MS;S) is maximized, in witch MS; S is the submatrix of M, restricted to rows and columns of S. This problem has a variety of applications including page ranking algorithms. The problem above is an NP-hard problem in the general case. However, there are several approximating and estimating algorithms for some special cases, which are interesting. Also, exactly solving the problem is NP-hard, even for some special matroids. As an example, one can consider the uniform matroids. Khachiyan introduced a greedy algorithm with dd approximation factor. Afterward, Di summa et al. proposed a better analysis of the algorithm. Civril and Magdon-Ismail, Nikolov introduced algorithm with a better approximation factor. There is another family of matroids investigated in this litreture which are partition matroids. Nikolov and Singh introduced a polynomial-time algorithm using geometric concave programming to estimate the value of maximum determinant under the constraint of partition matroid. They analyzed the expectation of algorithm, but can not conclude anything about variance. Ebrahimi, Straszak and Vishnoi introduced a new approach. They change the problem of finding a point in the cartesian product of simplices, in which a non-convex function is maximized. The improvement resulted from this approach is that in contradiction to the previous work of Nikolov and Singh a constraint of real stability and convexity of the function was omitted. Moreover, the result of the probability of obtaining a good result instead of the expectation value. I this thesis, we will discuss some previous works on these topics and present the case of partition and
    uniform matroid
  9. Keywords:
  10. Determinants ; Algorithm ; Real Stable Polynomial ; Concave Geometric Programming ; Partition Matroid ; Uniform Matroid

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