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...
Main Author: | |
---|---|
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 |
id |
ndltd-cmu.edu-oai-repository.cmu.edu-dissertations-1143 |
---|---|
record_format |
oai_dc |
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 |
_version_ |
1716709377745879040 |