1. More NP-completeness
Sipser 7.5 (pages )

2. NP’s hardest problems Definition 7.34: A language B is NP-complete if
B∈NP
A≤pB, for all A∈NP NP A1 A2
CLIQUE
SAT A3

3. Hamiltonian paths
HAMPATH = {<G,s,t> | ∃ Hamiltonian path from s to t}
Theorem 7.46: HAMPATH is NP-complete. s t

4. Hamiltonian paths
HAMPATH = {<G,s,t> | ∃ Hamiltonian path from s to t}
Theorem 7.46: HAMPATH is NP-complete. s t

5. Remember… HAMPATH∈NP N = "On input <G,s,t>:
Guess an orderings, p1, p2,..., pn, of the nodes of G
Check whether s = p1 and t = pn
For each i=1 to n-1, check whether (pi, pi+1) is an edge of G.
If any are not, reject. Otherwise, accept.”

6. 3SAT≤pHAMPATH NP A1 A2
HAMPATH
3SAT A3

7. Proof outline Given a boolean formula φ,
we convert it to a directed graph G such that φ has a valid truth assignment iff G has a Hamiltonian graph

8. 3SAT’s main features Choice:
Each variable has a choice between two truth values.
Consistency:
Different occurrences of the same variable have the same value.
Constraints:
Variable occurrences are organized into clauses that provide constraints that must be satisfied.
*We model each of these three features by a different a "gadget" in the graph G.

9. The choice gadget
Modeling variable xi

10. Zig-zagging and zag-zigging
Zig-zag (TRUE)
Zag-zig (FALSE)

11. The consistency gadget

12. Clauses
Modeling clause cj cj

13. The global structure

14. The constraint gadget
Modeling when clause cj contains xi

15. The constraint gadget
Modeling when clause cj contains xi

16. A situation that cannot occur

17. TSP is NP-complete TSP: Given n cities,
1, 2, ..., n, together with a nonnegative distance dij between any two cities,
find the shortest tour.

18. HAMPATH ≤p TSP NP A1 A2
TSP
HAMPATH A3

19. SUBSET-SUM is NP-complete
{<S,t> | S = {x1,…,xk} and,
for some {y1,…,yl} ⊆ S, Σ yi=t}
Why is SUBSET-SUM in NP?

20. 3SAT ≤p SUBSET-SUM

21. And…if that’s not enough
There are more than 3000 known
NP-complete problems!

22. Other types of complexity
Space complexity
Circuit complexity
Descriptive complexity
Randomized complexity
Quantum complexity …