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More NP-completeness Sipser 7.5 (pages )

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**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

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

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

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**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.”

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3SAT≤pHAMPATH NP A1 A2 HAMPATH 3SAT A3

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**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

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**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.

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The choice gadget Modeling variable xi

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**Zig-zagging and zag-zigging**

Zig-zag (TRUE) Zag-zig (FALSE)

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**The consistency gadget**

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Clauses Modeling clause cj cj

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The global structure

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The constraint gadget Modeling when clause cj contains xi

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The constraint gadget Modeling when clause cj contains xi

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**A situation that cannot occur**

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**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.

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HAMPATH ≤p TSP NP A1 A2 TSP HAMPATH A3

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**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?

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3SAT ≤p SUBSET-SUM

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**And…if that’s not enough**

There are more than 3000 known NP-complete problems!

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**Other types of complexity**

Space complexity Circuit complexity Descriptive complexity Randomized complexity Quantum complexity …

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