1. Lecture 4-4 #P-complete Problems

2. Is #P-complete is an extension of NP-complete?

3. Permanent

6. Perfect Matching
Each perfect matching corresponds to a permutation, also corresponds to a cycle cover. 1 1 1 4 2 2 3 3 4 4 2 3

7. #Perfect Matching
# perfect matching = permanent of adjacent matrix of bipartite graph. 1 1 2 2 3 3 4 4

8. Cycle Cover

9. Each permutation corresponds to a cycle cover.
Each permutation gives a term in permanent.

10. Equivalence between Perm and #CC

11. Theorem
PROOF
Called a Junction!

12. CONSTRUCTION FOR VARIABLE

13. No one comes to 1.
No one goes out from 4.

14. CONSTRUCTION FOR VARIABLE

15. CONSTRUCTION FOR VARIABLE

16. CONSTRUCTION FOR CLAUSE

17. ROUTE
GOOD ROUTE
CLAIM 1

18. PROOF OF CLAIM 1

19. CLAIM 2
PROOF OF CLAIM 2

20. CLAIM 3

21. CLAIM 3

22. CLAIM 3

23. CLAIM 3

24. CLAIM 3

25. CLAIM 3

26. CLAIM 3

27. CLAIM 3

28. CLAIM 4

29. 3-Perm
COROLLARY

30. THEOREM
PROOF

31. Chinese Remainder Theorem

32. Example: 4 persons a line; one is left.
5 persons a line; one position is left.

33. Example:
3 persons a line; 2 are left.
5 persons a line; 3 are left.

34. THEOREM
COROLLARY
PROOF OF THEOREM

39. Thanks, end.