Graph Partitioning and Semi-definite Programming Hierarchies

Graph partitioning is a fundamental optimization problem that has been intensively studied. Many graph partitioning formulations are important as building blocks for divide-and-conquer algorithms on graphs as well as to many applications such as VLSI layout, packet routing in distributed networks, c...

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Main Author: Sinop, Ali Kemal
Format: Others
Published: Research Showcase @ CMU 2012
Subjects:
Online Access:http://repository.cmu.edu/dissertations/145
http://repository.cmu.edu/cgi/viewcontent.cgi?article=1143&context=dissertations
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spelling ndltd-cmu.edu-oai-repository.cmu.edu-dissertations-11432014-07-24T15:35:47Z Graph Partitioning and Semi-definite Programming Hierarchies Sinop, Ali Kemal Graph partitioning is a fundamental optimization problem that has been intensively studied. Many graph partitioning formulations are important as building blocks for divide-and-conquer algorithms on graphs as well as to many applications such as VLSI layout, packet routing in distributed networks, clustering and image segmentation. Unfortunately such problems are notorious for the huge gap between known best known approximation algorithms and hardness of approximation results. In this thesis, we study approximation algorithms for graph partitioning problems using a strong hierarchy of relaxations based on semi-definite programming, called Lasserre Hierachy. Our main contribution in this thesis is a propagation based rounding framework for solutions arising from such relaxations. We present a novel connection between the quality of solutions it outputs and column based matrix reconstruction problem. As part of our work, we derive optimal bounds on the number of columns necessary together with efficient randomized and deterministic algorithms to find such columns. Using this framework, we derive approximation schemes for many graph partitioning problems with running times dependent on how fast the graph spectrum grows. Our final contribution is a fast SDP solver for this rounding framework: Even though SDP relaxation has nO(r) many variables, we achieve running times of the form 2O(r) poly(n) by only partially solving the relevant part of relaxation. In order to achieve this, we present a new ellipsoid algorithm that returns certificate of infeasibility. 2012-05-15T07:00:00Z text application/pdf http://repository.cmu.edu/dissertations/145 http://repository.cmu.edu/cgi/viewcontent.cgi?article=1143&context=dissertations Dissertations Research Showcase @ CMU approximation algorithms certificate of infeasibility column selection graph partitioning graph spectrum Lasserre hierarchy local rounding semi-definite programming strong duality Computer Sciences
collection NDLTD
format Others
sources NDLTD
topic approximation algorithms
certificate of infeasibility
column selection
graph partitioning
graph spectrum
Lasserre hierarchy
local rounding
semi-definite programming
strong duality
Computer Sciences
spellingShingle approximation algorithms
certificate of infeasibility
column selection
graph partitioning
graph spectrum
Lasserre hierarchy
local rounding
semi-definite programming
strong duality
Computer Sciences
Sinop, Ali Kemal
Graph Partitioning and Semi-definite Programming Hierarchies
description Graph partitioning is a fundamental optimization problem that has been intensively studied. Many graph partitioning formulations are important as building blocks for divide-and-conquer algorithms on graphs as well as to many applications such as VLSI layout, packet routing in distributed networks, clustering and image segmentation. Unfortunately such problems are notorious for the huge gap between known best known approximation algorithms and hardness of approximation results. In this thesis, we study approximation algorithms for graph partitioning problems using a strong hierarchy of relaxations based on semi-definite programming, called Lasserre Hierachy. Our main contribution in this thesis is a propagation based rounding framework for solutions arising from such relaxations. We present a novel connection between the quality of solutions it outputs and column based matrix reconstruction problem. As part of our work, we derive optimal bounds on the number of columns necessary together with efficient randomized and deterministic algorithms to find such columns. Using this framework, we derive approximation schemes for many graph partitioning problems with running times dependent on how fast the graph spectrum grows. Our final contribution is a fast SDP solver for this rounding framework: Even though SDP relaxation has nO(r) many variables, we achieve running times of the form 2O(r) poly(n) by only partially solving the relevant part of relaxation. In order to achieve this, we present a new ellipsoid algorithm that returns certificate of infeasibility.
author Sinop, Ali Kemal
author_facet Sinop, Ali Kemal
author_sort Sinop, Ali Kemal
title Graph Partitioning and Semi-definite Programming Hierarchies
title_short Graph Partitioning and Semi-definite Programming Hierarchies
title_full Graph Partitioning and Semi-definite Programming Hierarchies
title_fullStr Graph Partitioning and Semi-definite Programming Hierarchies
title_full_unstemmed Graph Partitioning and Semi-definite Programming Hierarchies
title_sort graph partitioning and semi-definite programming hierarchies
publisher Research Showcase @ CMU
publishDate 2012
url http://repository.cmu.edu/dissertations/145
http://repository.cmu.edu/cgi/viewcontent.cgi?article=1143&context=dissertations
work_keys_str_mv AT sinopalikemal graphpartitioningandsemidefiniteprogramminghierarchies
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