1. Prof. Busch - LSU1 Busch Complexity Lectures: More NP-complete Problems

2. Prof. Busch - LSU2 Theorem: If: Language is NP-complete Language is in NP is polynomial time reducible to Then: is NP-complete (proven in previous class)

3. Prof. Busch - LSU3 Using the previous theorem, we will prove that 2 problems are NP-complete: Vertex-Cover Hamiltonian-Path

4. Prof. Busch - LSU4 Vertex cover of a graph is a subset of nodes such that every edge in the graph touches one node in Vertex Cover S = red nodes Example:

5. Prof. Busch - LSU5 |S|=4Example: Size of vertex-cover is the number of nodes in the cover

6. Prof. Busch - LSU6 graph contains a vertex cover of size } VERTEX-COVER = { : Corresponding language: Example:

7. Prof. Busch - LSU7 Theorem: 1. VERTEX-COVER is in NP 2. We will reduce in polynomial time 3CNF-SAT to VERTEX-COVER Can be easily proven VERTEX-COVER is NP-complete Proof: (NP-complete)

8. Prof. Busch - LSU8 Let be a 3CNF formula with variables and clauses Example: Clause 1 Clause 2Clause 3

9. Prof. Busch - LSU9 Formula can be converted to a graph such that: is satisfied if and only if Contains a vertex cover of size

10. Prof. Busch - LSU10 Clause 1 Clause 2Clause 3 Clause 1 Clause 2Clause 3 Variable Gadgets Clause Gadgets nodes

11. Prof. Busch - LSU11 Clause 1 Clause 2Clause 3 Clause 1 Clause 2Clause 3

12. Prof. Busch - LSU12 If is satisfied, then contains a vertex cover of size First direction in proof:

13. Prof. Busch - LSU13 Satisfying assignment Example: We will show that contains a vertex cover of size

14. Prof. Busch - LSU14 Put every satisfying literal in the cover

15. Prof. Busch - LSU15 Select one satisfying literal in each clause gadget and include the remaining literals in the cover

16. Prof. Busch - LSU16 This is a vertex cover since every edge is adjacent to a chosen node

17. Prof. Busch - LSU17 Explanation for general case: Edges in variable gadgets are incident to at least one node in cover

18. Prof. Busch - LSU18 Edges in clause gadgets are incident to at least one node in cover, since two nodes are chosen in a clause gadget

19. Prof. Busch - LSU19 Every edge connecting variable gadgets and clause gadgets is one of three types: Type 1 Type 2 Type 3 All adjacent to nodes in cover

20. Prof. Busch - LSU20 If graph contains a vertex-cover of size then formula is satisfiable Second direction of proof:

21. Prof. Busch - LSU21 Example:

22. Prof. Busch - LSU22 exactly one literal in each variable gadget is chosen exactly two nodes in each clause gadget is chosen To include “internal’’ edges to gadgets, and satisfy chosen out of

23. Prof. Busch - LSU23 For the variable assignment choose the literals in the cover from variable gadgets

24. Prof. Busch - LSU24 since the respective literals satisfy the clauses is satisfied with

25. Prof. Busch - LSU25 Theorem: 1. HAMILTONIAN-PATH is in NP 2. We will reduce in polynomial time 3CNF-SAT to HAMILTONIAN-PATH Can be easily proven HAMILTONIAN-PATH is NP-complete Proof: (NP-complete)

26. Prof. Busch - LSU26 Gadget for variable the directions change

27. Prof. Busch - LSU27 Gadget for variable

28. Prof. Busch - LSU28

29. Prof. Busch - LSU29

30. Prof. Busch - LSU30