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More NP-completeness Sipser 7.5 (pages 283-294). CS 311 Fall 2008 2 NP’s hardest problems Definition 7.34: A language B is NP-complete if 1.B ∈ NP 2.A≤

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Presentation on theme: "More NP-completeness Sipser 7.5 (pages 283-294). CS 311 Fall 2008 2 NP’s hardest problems Definition 7.34: A language B is NP-complete if 1.B ∈ NP 2.A≤"— Presentation transcript:

1 More NP-completeness Sipser 7.5 (pages )

2 CS 311 Fall NP’s hardest problems Definition 7.34: A language B is NP-complete if 1.B ∈ NP 2.A≤ p B, for all A ∈ NP NP A1A1 SAT A3A3 A2A2 CLIQUE

3 CS 311 Fall Hamiltonian paths HAMPATH = { | ∃ Hamiltonian path from s to t} Theorem 7.46: HAMPATH is NP-complete. s t

4 CS 311 Fall Hamiltonian paths HAMPATH = { | ∃ Hamiltonian path from s to t} Theorem 7.46: HAMPATH is NP-complete. s t

5 CS 311 Fall Remember… HAMPATH ∈ NP N = "On input : 1.Guess an orderings, p 1, p 2,..., p n, of the nodes of G 2.Check whether s = p 1 and t = p n 3.For each i=1 to n-1, check whether (p i, p i+1 ) is an edge of G. If any are not, reject. Otherwise, accept.”

6 CS 311 Fall SAT≤ p HAMPATH NP A1A1 3SAT A3A3 A2A2 HAMPATH

7 CS 311 Fall 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 CS 311 Fall SAT ’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 satisified. *We model each of these three features by a different a "gadget" in the graph G.

9 CS 311 Fall The choice gadget Modeling variable x i

10 CS 311 Fall Zig-zagging and zag-zigging Zig-zag (TRUE)Zag-zig (FALSE)

11 CS 311 Fall The consistency gadget

12 CS 311 Fall Clauses Modeling clause c j cjcj

13 CS 311 Fall The global structure

14 CS 311 Fall The constraint gadget Modeling when clause c j contains x i

15 CS 311 Fall The constraint gadget Modeling when clause c j contains x i

16 CS 311 Fall A situation that cannot occur

17 CS 311 Fall TSP is NP-complete TSP : Given n cities, 1, 2,..., n, together with a nonnegative distance d ij between any two cities, find the shortest tour.

18 CS 311 Fall HAMPATH ≤ p TSP NP A1A1 HAMPATH A3A3 A2A2 TSP

19 CS 311 Fall SUBSET-SUM is NP-complete SUBSET-SUM= { | S = {x 1,…,x k } and, for some {y 1,…,y l } ⊆ S, y i =t} Why is SUBSET-SUM in NP?

20 CS 311 Fall SAT ≤ p SUBSET-SUM

21 CS 311 Fall And…if that’s not enough There are more than 3000 known NP-complete problems!

22 CS 311 Fall Other types of complexity Space complexity Circuit complexity Descriptive complexity Randomized complexity Quantum complexity …


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