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

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

3. Graph Matching Objective: matching a sample input graph to a database of known prototype graphs.

4. Graph Matching (cont’d) A real example: face identification

5. Graph Matching (cont’d) Key issues: (1) representative graph generation (a) facial graph representations (b) chemical graphs

6.  Maximum Common Subgraph (MCS)  Graph Edit Distance (GED) Enumeration procedures (for small graphs) Probabilistic models (MAP estimates) Binary Linear Programming (BLP) Graph Matching (cont’d) Key issues: (2) graph distance metrics

7. 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? Graph Edit Distance

8. Graph Edit Distance (cont’d) How to compute the distance between G 0 and G 1 ? Edit Grid

9. Isomorphisms of G 0 on the edit grid State Vectors Graph Edit Distance (cont’d) standard placement

10. Definition: (if the cost function c is a metric) Objective function: binary linear program (NP-hard!!!) Graph Edit Distance (Cont’d)

11. Lower bound: linear program (polynomial time) Upper bound: assignment problem (polynomial time) Graph Edit Distance (cont’d)

12. Edit Cost Selection Goal: suppose there is a set of prototype graphs {G i } i=1,…,N and we classify a sample graph G 0 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.

13. Edit Cost Selection (cont’d) Objective: minimize the variance of pairwise NN distances Define unit cost function, i.e., c(0,1)=1, c( ,  )=1, c( ,  )=0 Solve the BLP (with unit cost) and find the NN pair Construct H k,i = the number of i th edit operation for the k th NN pair Objective function: (convex optimization)

14. Experimental Results Chemical Graph Recognition

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

16. Experiments Results (cont’d) Optimal Edit Costs

17. A: GEDo B: GEDu C: MCS1 D: MCS2 Experiments Results (cont’d) Classification Results

18. 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. Conclusion