Graph Isoperimetry Problem Using Optimization Methods

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مساله ایزوپریمتری در گراف ها از دیدگاه بهینه سازی

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 46277 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Daneshgar, Amir
Abstract:
In this thesis, we study the mean graph isoperimetry problem using an optimization approach. The k-th isoperimetric constant of a graph is defined as the minimum of an objective function (p-norm of the vector consisting of normalized flow) over k-subpartitions of vertices. We note that the normalized cut problem can be formulated as a semidefinite programming problem and utilizing the relaxation methods for semidefinite programs, the problem can be solved in approximately polynomial time. Finally, we model the isoperimetry problem as an optimization problem with orthogonality constraints and utilizing Wen and Yin’s efficient method for finding local minima of the problem, we extract a subpartition as a solution.
We compare our approxamation algorithms with Daneshgar-Hajiabolhassan-
Javadi’s algorithm using some hard benchmarks and randomly generated instances of two dimensional clustering problem
Keywords:
Isoperimetric Constant ; Graphs ; Optimization ; Semidefinite Programming ; Normalized Cut Method