A%20Binary%20Linear%20Programming%20Formulation%20of%20the%20Graph%20Edit%20Distance

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A Binary Linear Programming Formulation of the
Graph Edit Distance
Authors Derek Justice Alfred Hero (PAMI 2006)
Presented by Shihao Ji Duke University Machine
Learning Group July 17, 2006
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Outline

• Introduction to Graph Matching
• Proposed Method (binary linear program)
• Experimental Results (chemical graph matching)

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Graph Matching

• Objective matching a sample input graph to a
  database of known prototype graphs.

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Graph Matching (contd)

• A real example face identification

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Graph Matching (contd)
Key issues (1) representative graph generation
(a) facial graph representations
(b) chemical graphs
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Graph Matching (contd)
Key issues (2) graph distance metrics

• Maximum Common Subgraph (MCS)
• Graph Edit Distance (GED)
• Enumeration procedures (for small
  graphs)
• Probabilistic models (MAP
  estimates)
• Binary Linear Programming (BLP)

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Graph Edit Distance

• Basic idea define graph edit operations (such as
  insertion or deletion or relabeling of a vertex)
  along with costs associated with each operation.
• The GED between two graphs is the cost associated
  with the least costly series of edit operations
  needed to make the two graph isomorphic.
• Key issues
• how to find the least costly series
  of edit operations?
• how to define edit costs?

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Graph Edit Distance (contd)

• How to compute the distance between G0 and G1?
• Edit Grid

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Graph Edit Distance (contd)

• Isomorphisms of G0 on the edit grid
• State Vectors

standard placement
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Graph Edit Distance (Contd)

• Definition (if the cost function c is a metric)
• Objective function binary linear program
  (NP-hard!!!)

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Graph Edit Distance (contd)

• Lower bound linear program (polynomial time)
• Upper bound assignment problem (polynomial time)

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Edit Cost Selection

• Goal suppose there is a set of prototype graphs
  Gi i1,,N and we classify a sample graph G0 by
  a nearest neighbor classifier in the metric space
  defined by the graph edit distance.
• Prior informaiton the prototypes should be
  roughly uniformly distributed in the metric space
  of graphs.
• Why it minimizes the worst case classification
  error since it equalizes the probability of error
  under a nearest neighbor classifier.

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Edit Cost Selection (contd)

• Objective minimize the variance of pairwise NN
  distances
• Define unit cost function, i.e., c(0,1)1,
  c(a,b)1, c(a,a)0
• Solve the BLP (with unit cost) and find the NN
  pair
• Construct Hk,i the number of ith edit operation
  for the kth NN pair
• Objective function (convex optimization)

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Experimental Results

• Chemical Graph Recognition

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Experiments Results (contd)
(a) original graph
1. edge edit 2. vertex deletion 3. vertex
insertion 4. vertex relabeling 5. random
(b) example perturbed graphs
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Experiments Results (contd)

• Optimal Edit Costs

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Experiments Results (contd)

• Classification Results

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Conclusion

• Present a binary linear programming formulation
  of the graph edit distance
• Offer a minimum variance method for choosing a
  cost metric
• Demonstrate the utility of the new method in the
  context of a chemical graph recognition.