1. Hamiltonian Path Problem This girl do this presentation. :) Teddy bear work along with me. Jeff help me understand the topic. Are they both cute?

2. Hamiltonian Path Problem in a Directed Graph Given a Directed Graph G, node s and t. Given a Directed Graph G, node s and t. s t Yes Hamiltonian Path is a path from s to t and go through every node in G exactly once. DHamPath DHamPath  NP-Complete

3. Step 0: DHamPath  NP Design poly-time algorithm Valid DHamPath (, P) which determines whether a path P is a Hamiltonian Path for. Therefore, DHamPath Therefore, DHamPath  NP Check if P starts at s and end at t. All nodes in G appear in P exactly once. None appear twice in P. Go through P and check if each edge of P is in the G.

4. DHamPath P easier ≤ poly DHamPath P easier  NP-Complete 3SAT Step 1: What to reduce it to

5. Step 2: What is What 3SAT Problem: Given an expression E: Does it have a satisfying assignment ? DHamPath: Given a directed graph G, s and t: Does it have a Hamiltonian Path from s to t? s t (x OR y OR z) AND (x OR w OR a) AND …

6. Given an oracle for DHAMPath, how can we solve 3SAT? Step 3: Direction of reduction & Code

7. 3SAT( Expression ) = DHAMPATH( )  Step5 Oracle will do it for us  Step 6 & 7 Step 8 & 9

8. Step 5: Instance Map  Expression Suppose we have K variables and L clauses in the expression. From C 1 to C L in Expression  From left to right in the diamond. For each clause C j, Use a pair of two adjacent nodes to represent it.... For each variable x i, we construct a variable diamond x i. Clause iClause i+1 xixi

9. Step 5: Instance Map Between each clause C j, use one spearator node to separate them.... xixi

10. Step 5: Instance Map xixi Since we have L clauses, we have 3L+1 nodes....

11. Step 5: Instance Map If we’d like to go through all nodes in variable diamond Xi EXACTLY ONCE. xixi... ONLY TWO WAYS.

12. Step 5: Instance Map We have K variable diamonds ……. x i+1 x i-1 xixi...

13. For each clause Ci,...... AND (x OR w OR a) AND … Step 5: Instance Map CiCi a CiCi... W x we use one node.

14. Step 3: Direction of reduction & Code 3SAT( Expression ) = DHAMPATH( )  Step5 Oracle will do it for us  Step 6 & 7 Step 8 & 9

15. Step 6: Solution Map Sol 3sat = SolutionMap( Sol DHam )... Xi  True Xi  False In variable diamond X i,

16. Step 3: Direction of reduction & Code 3SAT( Expression ) = DHAMPATH( )  Step5 Oracle will do it for us  Step 6 & 7 Step 8 & 9

17. Step 7: Valid to Valid Prove:A valid Sol DHam Sol 3sat is valid. Expression Yes That is, if we have a Hamiltonian Path, the expression will be true according to the mapping of variables.

18. If we have a Hamiltonian Path, it must go through every node once. So, it must go through every node in every variable diamond X i. xixi... Step 7: Valid to Valid

19. We have K diamonds …….... Step 7: Valid to Valid

20. Hamiltonian Path must go through each clause node Ci....... AND (x OR w OR a) AND … CiCi a CiCi... W x Step 7: Valid to Valid

21. We cannot have a Hamiltonian Path like that....... AND (x OR w OR a) AND … CiCi a CiCi... W x Step 7: Valid to Valid

22. If we have Hamiltonian Path like that:... CiCi Step 7: Valid to Valid

23. ... CiCi Step 7: Valid to Valid If we have Hamiltonian Path like that:

24. If there is a Hamiltonian Path, every clause nodes will belong to either one of two cases:... CiCi CiCi Step 7: Valid to Valid X i = True. xjxj xixi... AND (x i OR w OR a) AND … X j = False. CiCi... AND ( b OR x j OR a) AND … CiCi

25. Step 3: Direction of reduction & Code 3SAT( Expression ) = DHAMPATH( )  Step5 Oracle will do it for us  Step 6 & 7 Step 8 & 9

26. Sol DHam Sol 3sat Expression Yes If there is a satisfiable assignment, we will have a Hamiltonian Path in. Step 8 & 9: Reverse Solution Map

27. If there is a satisfiable assignment, For each variable diamond Xi, Step 8 & 9: Reverse Solution Map... X i = True. xjxj X j = False.... We have K variables. Go through them one by one from top to bottom. It is a Hamiltonian Path.

28. For each clause Ci,... AND (x OR w OR a) AND … Step 8 & 9: Reverse Solution Map at least one variable make it true. Choose one.

29. Step 8 & 9: Reverse Solution Map... CiCi CiCi... AND (x OR w OR a) AND … a x w

30. Step 3: Direction of reduction & Code 3SAT( Expression ) = DHAMPATH( )  Step5 Oracle will do it for us  Step 6 & 7 Step 8 & 9

31. Question?