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

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CS 311 Fall 2008 2 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

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CS 311 Fall 2008 3 Hamiltonian paths HAMPATH = { | ∃ Hamiltonian path from s to t} Theorem 7.46: HAMPATH is NP-complete. s t

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CS 311 Fall 2008 4 Hamiltonian paths HAMPATH = { | ∃ Hamiltonian path from s to t} Theorem 7.46: HAMPATH is NP-complete. s t

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CS 311 Fall 2008 5 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.”

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CS 311 Fall 2008 6 3SAT≤ p HAMPATH NP A1A1 3SAT A3A3 A2A2 HAMPATH

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CS 311 Fall 2008 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

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CS 311 Fall 2008 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 satisified. *We model each of these three features by a different a "gadget" in the graph G.

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CS 311 Fall 2008 9 The choice gadget Modeling variable x i

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CS 311 Fall 2008 10 Zig-zagging and zag-zigging Zig-zag (TRUE)Zag-zig (FALSE)

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CS 311 Fall 2008 11 The consistency gadget

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CS 311 Fall 2008 12 Clauses Modeling clause c j cjcj

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CS 311 Fall 2008 13 The global structure

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CS 311 Fall 2008 14 The constraint gadget Modeling when clause c j contains x i

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CS 311 Fall 2008 15 The constraint gadget Modeling when clause c j contains x i

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CS 311 Fall 2008 16 A situation that cannot occur

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CS 311 Fall 2008 17 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.

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CS 311 Fall 2008 18 HAMPATH ≤ p TSP NP A1A1 HAMPATH A3A3 A2A2 TSP

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CS 311 Fall 2008 19 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?

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CS 311 Fall 2008 20 3SAT ≤ p SUBSET-SUM

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CS 311 Fall 2008 21 And…if that’s not enough There are more than 3000 known NP-complete problems! http://en.wikipedia.org/wiki/List_of_NP-complete_problems

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CS 311 Fall 2008 22 Other types of complexity Space complexity Circuit complexity Descriptive complexity Randomized complexity Quantum complexity …

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