Loading...

Isoperimetric Problems on Graphs

Javadi Jourtani, Ramin | 2011

1388 Viewed
  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 42235 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Daneshgar, Amir
  7. Abstract:
  8. This thesis is concerned with studying several aspects of the generalized isoperimetric problems on weighted graphs. In this regard, as a generalization of well-known Cheeger constant, we define k-th isoperimetric constant as the minimum of normalized outgoing flow over all k-subpartitions (disjoint subsets) of the vertex set. Also, we investigate basic properties of these parameters and particularly we prove a Federer-Fleming-type theorem. We then introduce a generalized Cheeger conjecture which is a generalization of celebrated Cheeger inequality and relates defined isoperimetric constants to higher eigenvalues of Laplace operator. In order to find a partial proof for this conjecture, we deploy two approaches by applying nodal domains and Dirichlet connectivity spectrum and prove that a generalized Cheeger conjecture is valid with a universal constant, for all Laplacians whose underlying graphs are trees or cycle. Moreover, as a by-product, we characterize spectral decomposition of Laplacians on cycles and describe the shape of their eigenfunctions. Finally, in direction of studying computational aspects of generalized isoperimetric problems, we obtain several hardness results and introduce some novel efficient algorithms for computing isoperimetric constants on weighted trees. Among these results, we show that for weighted trees, the decision problem for the k-th maximum isoperimetric number can be solved in linear time, while the decision problems for the k-th mean isoperimetric number and the k-th max/mean minimum normalized cuts are NP-complete. Furthermore, in direction to explore parametrized complexity of the problems, we investigate the case of fixed k and for weighted trees, we present some algorithms to determine the k- th max/mean isoperimetric numbers and the k-th max minimum normalized cuts.
  9. Keywords:
  10. Computational Complexity ; Clustering ; Weighted graphs ; Isoperimetric Constant ; Generalized Cheeger Inequalities ; Isoperimetric Spectrum ; Dirichlet Connectivity Spectrum

 Digital Object List

 Bookmark

...see more