Ant Based Optimization for Multiway Graph Partition

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Ant Based Optimization for Multiway Graph
Partition

• Ezra Nugroho
• February 28th, 2005

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Outline

• Problem Definition
• Ant Based Optimization (ABO) and Ant
  Colony Optimization (ACO)
• ABO for Multiway Graph Partitioning (ABMGP)
• Conclusion

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Outline

• Problem Definition
• Ant Based Optimization (ABO) and Ant
  Colony Optimization (ACO)
• ABO for Multiway Graph Partitioning (ABMGP)
• Conclusion

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Problem Multiway Graph Partition

• Input
• G (V, E) an undirected graph
• k an integer from 2 to V
• Output
• S1, , Sk k disjointed subsets of V such that
• Si n Sj ø for i ? j
• U Si V
• Si ? ø for i 1, , k
• Sl - Ss 1, Sl the largest subset, Ss the
  smallest one
• Goal
• The crossing edges between subsets are minimized

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Problem Further Definitions

• Other name Graph k-way Partition Problem
• (a, k)-Partition Problem, also called Ratio Cut
  Problem
• Allows unbalanced partition
• Sl a V , Sl the largest subgraph
• 2-way Partition Problem is often called the
  Bisection Problem
• The size of the set of crossing edges is called
  the cutsize

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2-way Partition
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2-way Partition
Cutsize 6
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2-way Partition
Cutsize 6
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2-way Partition
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2-way Partition
Cutsize 2
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Problem Complexity

• Multiway Graph Partition Problem is NP-hard
• Approximating good cut size for general graph is
  NP-hard
• The bisection problem is NP-hard, even for
  bipartite graphs
• Exact polynomial algorithm exists for trees and
  planar graphs with O(log n) optimal cutsize

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Problem Complexity (cont)

• Let V n, the size of search space for
  the k-way partition problem is
• Grows very fast as n increases

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Problem Applications

• Circuit Design
• Scheduling for parallel processing
• Initial Process for Divide and Conquer algorithms
• General Layout Network design, Routing, etc.

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Problem Existing Algorithms

• KernighanLin algorithm for bisection
• Recursive KernighanLin
• Pairwise KernighanLin
• Cyclic k-way Partitioning Algorithm
• Other Heuristics and Hybrid Algorithms
• Genetic Algorithms
• Simulated Annealing
• Tabu Search
• Hybrid Ant Based Algorithm

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Outline

• Problem Definition
• Ant Based Optimization (ABO) and Ant
  Colony Optimization (ACO)
• ABO for Multiway Graph Partitioning (ABMGP)
• Conclusion

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ABO/ACO Similarities

• Agent-based Optimization
• Inspired by ant colonies
• Inspired by other animals ? animats
• Animats use pheromones to send signals to others
• Good in identifying possible good solution
• Cannot fine-tune solutions

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Ant Based Optimization (ABO)

• Multi Agents, one solution
• Animats move in parallel
• Problem domain clustering, landscape
  identification

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Ant Colony Optimization (ACO)

• One agent, one solution
• Usually Animats move serially
• Problem domain path problems

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ABO/ACO Applications

• ABO
• k-Cardinality Tree problem
• Clique problem
• Bisection problem
• Etc.
• ACO
• TSP
• Quadratic Assignment
• Routing
• Etc.

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Outline

• Problem Definition
• Ant Based Optimization (ABO) and Ant
  Colony Optimization (ACO)
• ABO for Multiway Graph Partitioning (ABMGP)
• Conclusion

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ABMGP Algorithm Overview

• Phase 1
• 20 rounds of finding good candidate solutions and
  locally optimize them
• Phase 2
• Fully optimize the best candidate solution

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ABMGP Algorithm Detail

• Phase 1
• 20 rounds of finding good candidate solutions and
  locally optimize them
• Phase 2
• Fully optimize the best candidate solution

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ABMGP Algorithm Detail

• Phase 1
• Create initial k-way partition
• for round 1 to 20
• for iteration 1 to max
• Place animats in vertices
• if (progressing)
• Run animats
• else
• Perform random jumps (jolt)
• end if-else
• Rebalance with certain probability
• Record good partition
• end
• Push the best solution from this round closer to
  an optimum
• end
• Phase 2
• Fully optimize the absolute best solution

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ABMGP Algorithm Detail

• Create initial k-way partition
• for round 1 to 20
• for iteration 1 to max
• Place animats in vertices
• if (progressing)
• Run animats
• else
• Perform random jumps (jolt)
• end if-else
• Rebalance with certain probability
• Record good partition
• end
• Push the best solution from this round closer to
  an optimum
• end
• Fully optimize the absolute best solution

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ABMGP Animat Activation

• Ideas
• We use k colonies of territorial animats, each
  territory is a subgraph
• Animats try to enlarge their territory by
  attacking other animats in different vertices
• Animats use pheromones to communicate with each
  other to encourage collaborative decisions
  defending vertices, attacking vertices
• An Animat captures a vertex v if it defeats the
  last animat in v

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ABMGP Animat Activation
No Way!
We want that vertex!
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ABMGP Animat Activation
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ABMGP Animat Activation
Attacker wins, defender is removed
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ABMGP Animat Activation
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ABMGP Animat Activation
Attacker looses, it waits for next iteration
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ABMGP Animat Activation
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ABMGP Animat Activation
Attacker wins, defender is removed
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ABMGP Animat Activation
Attacker wins, defender is removed
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ABMGP Animat Activation
vertex is captured
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ABMGP Animat Activation
The colonies are updated
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ABMGP Algorithm Detail

• Create initial k-way partition
• for round 1 to 20
• for iteration 1 to max
• Place animats in vertices
• if (progressing)
• Run animats
• else
• Perform random jumps (jolt)
• end if-else
• Rebalance with certain probability
• Record good partition
• end
• Push the best solution from this round closer to
  an optimum
• end
• Fully optimize the absolute best solution

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ABMGP Jolts

• Pairs of vertices from different colonies are
  randomly chosen
• Swap the owner
• Adjusts the pheromone levels

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ABMGP Algorithm Detail

• Create initial k-way partition
• for round 1 to 20
• for iteration 1 to max
• Place animats in vertices
• if (progressing)
• Run animats
• else
• Perform random jumps (jolt)
• end if-else
• Rebalance with certain probability
• Record good partition
• end
• Push the best solution from this round closer to
  an optimum
• end
• Fully optimize the absolute best solution

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ABMGP Algorithm Detail

• KernighanLin algorithm for Bisection
• Starting with a bisection (A, B)
• Cut size is reduced by swapping pairs of vertices
• Identify pairs of vertices (a, b) that reduce the
  cut size the most
• Vertices in these pairs are swapped
• Repeat

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2-way Partition
Cutsize 6
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2-way Partition
A
A
(A, A) gain 0
Cutsize 6
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2-way Partition
B
A
A
(A, A) gain 0 (B, B) gain 4
B
Cutsize 6
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2-way Partition
B
A
A
B
Cutsize 2
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ABMG Algorithm Detail

• Weaker derivation of Pairwise KernighanLin
  Algorithm
• Cut size between subgraph pairs are computed
• Pairs are sorted according to their cut sizes
• Run KL for bisection algorithm for each pair
• Some vertex-swaps are considered

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ABMGP Algorithm Detail

• Phase 1
• 20 rounds of finding good candidate solutions and
  locally optimize them
• Phase 2
• Fully optimize the best candidate solution

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ABMG Algorithm Detail

• Full Pairwise KernighanLin Algorithm
• Cut size between subgraph pairs are computed
• Pairs are sorted according to their cut sizes
• Run KL for bisection algorithm for each pair
• All possible vertex-swaps are considered

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ABMGP Test Bed

• 40 Graphs from 8 different classes
• Size from 100 5252
• Average degree from 2 to 36
• Random and geometric

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ABMGP Implementation

• ABMGP is implemented in C
• 3.06 Ghz Xeon Processor
• 2 G of RAM
• Linux
• 100 runs for each graph

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ABMGP Result for bisection
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ABMGP Result 4-way Partition
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ABMGP Result for 8-way Partition
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ABMGP Result for 32-way Partition
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Outline

• Problem Definition
• Ant Based Optimization (ABO) and Ant
  Colony Optimization (ACO)
• ABO for Multiway Graph Partitioning (ABMGP)
• Conclusion

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Conclusion

• ABMGP is given
• Hybrid of ABO and Pairwise KL
• Generally results are competitive
• ABMGP does not excel in every direction
• ABMGP matches many best known solution
• ABMGP also finds several new best known