balanced graph partitioning konstantin andreev harald räcke
Post on 19-Dec-2015
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Balanced Graph Partitioning
Konstantin Andreev
Harald Räcke
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k - balanced graph partitioning
G=(V,E)
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Motivation
Parallel Computing
VLSI design
Sparse Linear System Solving
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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
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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.
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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
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3-Partition
A
a1 a2 a3 a4 a5 a6 a7 a8 a9
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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.
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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.
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Hierarchical Decomposition
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Decomposition Tree - T
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Partitions induced by T
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Partitions induced by T
O(log n/)
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Approximation ratio
Leighton-Rao’s
(, 1-) – separation
algorithm
Height of
the tree
decomposition
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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
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Decomposition Tree Pruning
T1T2
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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
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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.
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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.
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Thank you!