# More NP-completeness Sipser 7.5 (pages 283-294).

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More NP-completeness Sipser 7.5 (pages )

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

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

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

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.”

3SAT≤pHAMPATH NP A1 A2 HAMPATH 3SAT A3

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

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.

The choice gadget Modeling variable xi

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

Clauses Modeling clause cj cj

The global structure

The constraint gadget Modeling when clause cj contains xi

The constraint gadget Modeling when clause cj contains xi

A situation that cannot occur

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.

HAMPATH ≤p TSP NP A1 A2 TSP HAMPATH A3

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?

3SAT ≤p SUBSET-SUM

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

Other types of complexity
Space complexity Circuit complexity Descriptive complexity Randomized complexity Quantum complexity