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5th European Conference on Evolutionary
Computation in Combinatorial Optimization
Problems EvoCOP 2005
ANT Algorithm for the Graph Matching Problem
Olfa Sammoud, Christine Solnon Khaled Ghédira
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Outline

• Graph Matching Problem
• ANT- Graph Matching (ANT-GM)
• Experimental study
• Conclusion

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Part I Graph Matching Problem

• Similarity
• Graph Matching Problem
• Existing Graph Matching Problems
• A generic graph similarity measure

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Similarity
Measuring the Similarity of Objects ? Identify
and quantify their common features and their
differences
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Graph Matching Problem
When OBJECTS are modeled by ?
GRAPHS
Components ?
Vertices
Relations between Components ? Edges
Features ? Labels
Similarity ??
Graph Matching Problem
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Existing Graph Matching Problems

• Class 1 Exact graph matching (Isomorphism)

• Searching for the matching which preserves all
  vertex and all edge features.
• To prove graph equivalence or graph inclusion

Sub-graph isomorphism

• Usually graphs are not identical (noise,
  over-segmentation...)

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• Class 2 Error-Tolerant Graph Matching

? Searching for the best matching which
preserves a maximum number of vertex and edge
features.
m ? (a,1), (b,2), (c,3), (d,4), (f,5)

• The role of one vertex may be played by a set of
  vertices
• Univalent mapping (associating each vertex with
  at most one vertex in the other graph) becomes
  insufficient.

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• Class 3 Multivalent Graph Matching

• Searching for the best matching which
  preserves a maximum number of features
• Possibility to map one vertex to a set of
  vertices

m ? (a,1), (b,2), (c,3), (d,4), (f,5), (e,5)
Idea Similarity of two graphs depends on their
common features, Graph features ?
Graph descriptor (descr)
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Graph similarity
Graph Features ? Graph descriptor (descr)
Descr(G) (a,beam), (b,beam), (c,beam),
(d,beam),
(a,I), (b,I), (c,I), (d,I),
(a,b,next to), (b,c,next to),
(c,d,next to), (e,f,next to),
(e,wall), (f,wall),
(a,e,on), (b,e,on), (c,f,on), (d,f,on)
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A Generic similarity measure Champin
Solnon 03
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Example
m ?

(a,1),
(b,2),
(e,5),
(c,3),
(d,4),
(f,5)
descr(G1) ? descr(G2) ? (a,beam), (b,beam),
(c,beam), (d,beam), (a,I), (b,I), (c,I), (d,I),
(1,beam), (2,beam), (3,beam), (4,beam), (a,U),
(2,U), (3,U), (4,U), (e,wall), (f,wall),
(5,wall), (a,b,next to), (b,c,next to), (c,d,next
to), (a,e,on), (b,e,on), (c,f,on), (d,f,on),
(1,2,next to), (2,3,next to), (3,4,next to),
(1,5,on), (2,5,on), (3,5,on), (4,5,on) ,
(e,f,next to) ? 34 Features
descr(G1) ?m descr(G2) ? (a,beam), (1,beam),
(b,beam), (2,beam), (a,b,next to), (1,2,next to),
(e,wall), (5,wall), (a,e,on), (b,e,on), (1,5,on),
(2,5,on) , (c,beam), (3,beam), (b,c,next to),
(2,3,next to), (d,beam), (4,beam), (c,d,next to),
(3,4,next to),(f,wall), (c,f,on), (d,f,on),
(3,5,on), (4,5,on) ? 25 common features 1
split
descr(G1) ?m descr(G2) ? (a,beam), (1,beam),
(b,beam), (2,beam), (a,b,next to), (1,2,next to),
(e,wall), (5,wall), (a,e,on), (b,e,on), (1,5,on),
(2,5,on)
descr(G1) ?m descr(G2) ? (a,beam), (1,beam)
descr(G1) ?m descr(G2) ? (a,beam), (1,beam),
(b,beam), (2,beam), (a,b,next to), (1,2,next to)
descr(G1) ?m descr(G2) ?
If we define f and g as cardinality functions ?
simm(G1,G2)(25-1)/340.70
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Part II ANT-GM Ant-Graph Matching

• ANT-GM basic idea
• ANT-GM algorithm
• ? Define construction graph
• ? Construction of mapping by an ant
• ? Pheromone update

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ANT-GM basic idea
Use Ant Colony Optimization (ACO) to improve the
quality of the constructed matchings ? Use
Pheromone to keep track of the most promising
components when constructing new matchings

• Problem to solve ? search for a Best path
  in a graph called Construction Graph

• Use artificial ants ? search for Good paths
• ? Probabilistic construction of paths guided by
  pheromone
• ? Pheromone laying on best paths
• ? Pheromone evaporation

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ANT-GM algorithm

• Define a construction graph
• Initialize pheromone trails of the construction
  graph to ?0
• Repeat
• Each ant constructs a mapping
• Update pheromone trails
• Until Optimal mapping is found or Max Cycle is
  reached

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Construction graph

• For measuring the similarity of two graphs G1
  (V1,E1) and G2 (V2,E2), we define the
  construction graph G(V,E)

• Complete and non-directed graph
• V V1 ? V2
• ? (u,u) selected by an ant ? she prefers to
  match vertex u of G1 with the vertex u of G2

• ?lt(u,u), (v,v)gt ? pheromone trail between
  (u,u) and (v,v)
• ? learnt desirability of matching together u
  with u and v with v

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Mapping construction by an ant
m? ?
Iteratively add a couple (u,u) to m

• cand ? (u,u) ? V - m /
• simm ? (u,u)gt simm or look_ahead (u,u)gt 0

• Choose (u,u) ? cand with a probability pm(u,u)
  

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?, ?1 and ?2 determine the relative importance
of the 3 factors
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Pheromone update

• Evaporation
• For each edge lt(u,u),(v,v)gt in E,
• ?lt(u,u), (v,v)gt ? ? . ?lt(u,u), (v,v)gt
• Where ? is the pheromone evaporation rate (0 ? ?
  ? 1)

• Reward
• let mk the best matching built during
  the current cycle
• mbest the best matching built since the
  beginning of run
• For each edge lt(u,u),(v,v)gt in mk
• ?lt(u,u), (v,v)gt ? 1 /1 sim(mbest)-sim(mk)

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Part III Experimental Study

• Parameters influence
• Test suite 1 sub-graph isomorphism problems
• Test suite 2 multivalent matching problems

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Quality of the best matching
Parameters influence

• Pheromone weight ? ? 1
• Evaporation rate ? ? 0.01
• Max number of cycles? Maxcycle 1000
• Heuristic Inf. weights ? ?1 8 ?2 3
• Number of ants ? NbAnts 10
• Initial Pheromone ? ?0 6

without pheromone (alpha0, rho1) small
influence of pheromone (alpha1,
rho0.01) strong influence of pheromone (alpha2,
rho0.02)
Number of cycles (logscale)
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Test suite 1 sub-graph isomorphism

• Considered algorithms
• Our ACO algorithm ANT-GM
• Greedy Search GS Champin Solnon 03
• Reactive Tabu Search RTS Sorlin Solnon 05
• We fix a same number of moves for each algorithm
  where
• a move in ANT-GM/GS ? vertex couple adding
• a move in RTS ? vertex couple
  adding/deleting

• 11 Sub-graph isomorphism problems Foggia al
  01
• Each problem, we consider the 30 first
  instances,
• so 330 instances ( 11 benchs 30 instances) are
  treated
• Each instance is run 20 times,
• so, 6600 runs ( 330 instances 20 runs) are
  done

f and g as cardinality functions
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• Comparison criteria
• Global Success Rate (GSR) percentage of
  successful runs over the 6600 runs
• Instance Success Rate (ISR) percentage of
  instance that have been solved at least once over
  the 20 runs over the 33O instances
• Number of moves (Mv) (average on successful
  runs)
• time (t in second) (average on successful runs)

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Results on sub-graph isomorphism
problem Foggia al 01

Bench name ANT-GM ANT-GM ANT-GM ANT-GM GS Champin Solnon 03 GS Champin Solnon 03 GS Champin Solnon 03 GS Champin Solnon 03 RTS Sorlin Solnon 05 RTS Sorlin Solnon 05 RTS Sorlin Solnon 05 RTS Sorlin Solnon 05
Bench name GSR ISR Mv t GSR ISR Mv t GSR ISR Mv t
si2r001s100 76.8 86.7 40492 33.2 33.3 33.3 89 0.2 67.5 100 9758 6.6
si2r001s80 93.3 100 42240 10.1 33.3 33.3 37 0.0 90.0 100 5585 2.4
si2r001s60 99.7 100 22164 2.8 46.7 46.7 15 0.0 99.2 100 1590 0.4
si4r001s80 81.3 90.0 110818 44.1 23.3 23.3 507 0.5 85.7 100 8292 7.5
si4r001s60 99.2 100 44539 9.0 40.0 40.0 49 0.1 93.2 100 5066 2.5
si4r001s40 100 100 48634 0.7 53.3 53.3 41 0.0 99.7 100 1759 0.4
si4r001s20 100 100 166 0.0 83.3 83.3 9 0.0 100 100 219 0.0
si4r005s40 89.7 96.7 34996 4.4 6.7 6.7 67 0.0 88.0 96.7 4647 1.0
si6r001s60 99.7 100 79738 21.0 63.3 63.3 110 0.1 94.5 100 6964 5.2
si6r001s40 100 100 16547 1.9 86.7 86.7 44 0.0 98.3 100 31.0 1.0
si6r001s20 100 100 352 0.0 93.3 93.3 24 0.0 100 100 266 0.0
Average 94.5 97.6 36424 11.6 51.2 51.2 89 0.1 92.4 99.7 4295 2.45
Bench name ANT-GM ANT-GM ANT-GM ANT-GM RTS Sorlin Solnon 05 RTS Sorlin Solnon 05 RTS Sorlin Solnon 05 RTS Sorlin Solnon 05
Bench name GSR ISR Mv t GSR ISR Mv t
Average 94.5 97.6 36424 11.6 92.4 99.7 4295 2.45
Results obtained by ANT-GM and RTS are rather
complementary
Global Success Rate ANT-GM 94.5 ? 363
runs/6600 have failed RTS 92.4 ? 501
runs/6600 have failed
Instance Success Rate ANT-GM 97.6 ? 7
instances/330 not solved RTS 99.7 ? 1
instance/330 not solved
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Test Suite 2 results on multivalent graph
matching Problems
Problem name ANT-GM ANT-GM ANT-GM RTS RTS RTS
Problem name Similarity degree Mv t Similarity degree Mv t
hom-v20-e60 0.795 303167 30.9 0.798 17747 2.2
hom-v30-e90 0.863 512746 155.0 0.865 14187 4.4
hom-v40-e120 0.885 685155 477.9 0.895 24801 13.7
hom-v45-e135 0.895 717767 709.5 0.904 60085 40.5
hom-v50-e150 0.804 847699 1075.6 0.913 53922 47.9
Average 0.848 613307 489.8 0.875 34149 21.7
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Conclusion

• ACO improves the quality of matchings
• Results obtained by ANT-GM and RTS are rather
  complementary
• For multivalent graph matching, ANT-GM is
  outperformed by RTS

• Further Work Integrate local search within
  ANT-GM

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Fifth European Conference on Evolutionary
Computation of Combinatorial Optimization
Problems EvoCOP 2005
ANT Algorithm for the Graph Matching Problem
Olfa Sammoud, Christine Solnon Khaled Ghédira