More NP-completeness

Sipser 7.5 (pages 283-294)

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NP’s hardest problems

• Definition 7.34: A language B is NP-complete if1. B NP∈2. A≤pB, for all A NP∈

NP

A1

SATA3

A2

CLIQUE

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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>: 1. Guess an orderings, p1, p2,..., pn, of the

nodes of G2. Check whether s = p1 and t = pn

3. 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

3SATA3

A2

HAMPATH

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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

HAMPATHA3

A2

TSP

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SUBSET-SUM is NP-complete

• SUBSET-SUM=

{<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!

http://en.wikipedia.org/wiki/List_of_NP-complete_problems

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Other types of complexity

• Space complexity• Circuit complexity• Descriptive complexity• Randomized complexity• Quantum complexity• …