balanced graph partitioning l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Balanced Graph Partitioning PowerPoint Presentation
Download Presentation
Balanced Graph Partitioning

Loading in 2 Seconds...

play fullscreen
1 / 20

Balanced Graph Partitioning - PowerPoint PPT Presentation


  • 387 Views
  • Uploaded on

Balanced Graph Partitioning. Konstantin Andreev Harald R ä cke. k - balanced graph partitioning. G=(V,E). Motivation. Parallel Computing VLSI design Sparse Linear System Solving. Problem Definition.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Balanced Graph Partitioning' - liam


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
balanced graph partitioning

Balanced Graph Partitioning

Konstantin Andreev

Harald Räcke

motivation
Motivation
  • Parallel Computing
  • VLSI design
  • Sparse Linear System Solving
problem definition
Problem Definition
  • For a graph G=(V,E) we call a partitioning P, -balanced if V is partitioned into k disjoint subsets each containing at most vertices.
  • Denote with cost(P) the capacity of edges cut by the partitioning P
  • Find the minimum cost

-balanced graph partitioning

related work
Related Work
  • Even et al. showed that any (k,)-balanced partitioning with  > 2 can be reduced to a (k’,1+) where · 1.
  • Furthermore they gave a O(log n) bicriteria approximation for the (k, 2)-balanced partitioning problem.
  • Feige and Krauthgamer gave a O(log2 n) approximation for minimum bisection, i.e. the (2,1)-balanced graph part.
our results
Our Results
  • We prove that (k,1)-balanced part. is inapproximable within any finite constant unless P=NP
  • We present a O(log2 n/4) factor bicriteria approximation for the (k,1+)-balanced graph part. problem
3 partition
3-Partition

A

a1 a2 a3 a4 a5 a6 a7 a8 a9

hardness result
Hardness Result
  • 3-Partition problem: Given a1,a2, ..,a3k integers, a threshold A s.t. A/4<ai<A/2 and  ai = kA, decide if the numbers can be partitioned into triples so that every triple sums up to exactly A.
  • This problem is strongly NP-complete, i.e. it is NP-complete even if all ai and A are polynomialy bounded.
reduction
Reduction
  • Assume we can approximate (k,1)-balanced graph part. within a finite factor.
  • For an instance of 3-Partition construct the graph G so that for every ai we have a clique of size ai and all of them are disconnected.
  • 3-Partition can be solved if the (k,1)-balanced graph part. algorithm can differentiate between not cutting edges and cutting at least one edge.
approximation ratio
Leighton-Rao’s

(, 1-) – separation

algorithm

Height of

the tree

decomposition

Approximation ratio
decomposition tree pruning
Decomposition Tree Pruning
  • Observation: Tree nodes that have less than vertices or more than . graph vertices in them do not have to be considered.
  • Thus we are left with a forest of sub-trees all which have constant height
dynamic programming algorithm
Dynamic Programming Algorithm
  • Let g1, ..,gt denote the number of sets of different sizes that are used in the clustering of T1, .., Ti -1.
  • If g1, ..,gt is infeasible then
  • Otherwise
running time
Running Time
  • Dynamic programming table has

entries.

  • To decide whether g1, ..,gt is feasible takes time.
  • To compute the minimum in the recursion over all partitionings of Ti takes constant time.
  • The separation algorithm takes time.
future work
Future Work
  • Solve the generalized problem when different partitions are required to have different sizes.
  • Improve the dependence on 1/ of the approximation ratio or the running time.
  • Improve the approximation ratio.