Balanced Graph Partitioning

Konstantin Andreev

Harald Räcke

2

k - balanced graph partitioning

G=(V,E)

3

Motivation

Parallel Computing

VLSI design

Sparse Linear System Solving

4

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

5

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.

6

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

7

3-Partition

A

a1 a2 a3 a4 a5 a6 a7 a8 a9

8

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.

9

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.

10

Hierarchical Decomposition

11

Decomposition Tree - T

12

Partitions induced by T

13

Partitions induced by T

O(log n/)

14

Approximation ratio

Leighton-Rao’s

(, 1-) – separation

algorithm

Height of

the tree

decomposition

15

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

16

Decomposition Tree Pruning

T1T2

17

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

18

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.

19

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.

20

Thank you!