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Exact and Approximation Algorithms for the Isoperimetry Problem

Alimi, Morteza | 2023

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 56217 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Daneshgar, Amir
  7. Abstract:
  8. Graph partitioning and clustering are among the most important problems in computer science and engineering with a wide range of applications, to which a large number of research projects have attended. The normalized cut ratio is one of the best metrics for graph partitioning, giving rise to the ``Isoperimetry'' problem, which is, however, a challenging problem in its various forms. In this dissertation, we investigate the Isoperimetry problem. %, focusing on the Mean version. We prove that the Mean Isoperimetry problem on edge-weighted trees is solvable in strongly polynomial time. The algorithm can accomodate a wide range of constraints on the problem in the connected regime, including having control on the number of outliers and bounds on the sizes of the parts. More generally, we show that on edge-weighted graphs with bounded treewidth, the problem is solvable in strongly polynomial time. Also, we present an $O(\log n)$-approximation algorithm for the problem on edge-weighted graphs. We also study the Max Isoperimetry problem on the same classes of graphs. We present an algorithm for this problem on edge-weighted graphs which can handle a wide variety of extra constraints in the connected case, specifically bounds on the sizes of the parts containing each vertex, and having penalties for outliers. We also show that on bounded treewidth unweighted graphs, this problem is polynomial-time solvable. For the case of fixed $k$, we show $O(\log n)$-approximability of the Max Isoperimetry problem on edge-weighted graphs. As for the general (norm $p$) version of the Isoperimetry problem, we prove that the problem is polynomial-time solvable on unweighted trees. Furthermore, we propose a method for approximating the problem on edge-weighted trees. As an application, we show how the $O(\log n)$-approximation algorithm can be utilized to give an $O(\log n)$-approximation algorithm for a general version of the well-known $k$-means clustering problem.
  9. Keywords:
  10. Graph Partitioning ; Clustering ; Normalized Cut Method ; K-means Clustering ; Approximate Algorithm ; Bounded-Treewidth Graphs ; Cut Sparsifier ; Isoperimetry

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