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 G

2. 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 satisified.

*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, ⊆

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• …