More NP-completeness Sipser 7.5 (pages 283-294). CS 311 Fall 2008 2 NP’s hardest problems...
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Transcript of More NP-completeness Sipser 7.5 (pages 283-294). CS 311 Fall 2008 2 NP’s hardest problems...
More NP-completeness
Sipser 7.5 (pages 283-294)
CS 311Fall 2008
2
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 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• …