Balanced Graph Partitioning

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# Balanced Graph Partitioning - PowerPoint PPT Presentation

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.

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## Balanced Graph Partitioning

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### Balanced Graph Partitioning

Konstantin Andreev

Harald Räcke

Motivation
• Parallel Computing
• VLSI design
• Sparse Linear System Solving
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
• 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
• 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

A

a1 a2 a3 a4 a5 a6 a7 a8 a9

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
• 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.
Leighton-Rao’s

(, 1-) – separation

algorithm

Height of

the tree

decomposition

Approximation ratio
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
• 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
• 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
• 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.