lecture 25 np class. p = ? np = ? pspace they are central problems in computational complexity
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Lecture 25 NP Class
P = ? NP = ? PSPACE
• They are central problems in computational complexity.
If P = NP, then
NP-completeP
Ladner Theorem
• If NP ≠ P, then there exists a set A lying -between P and NP-complete class, i.e., A is in NP, but not in P and not being NP-compete.
How to prove a decision problem belonging to NP?
How to design a polynomial-time nondeterministic algorithm?
Hamiltonian Cycle
• Given a graph G, does G contain a Hamiltonian cycle?
• Hamiltonian cycle is a cycle passing every
vertex exactly once.
Nondeterministic Algorithm
• Guess a permutation of all vertices.• Check whether this permutation gives a
cycle. If yes, then algorithm halts.
What is the running time?
Minimum Spanning Tree
• Given an edge-weighted graph G, find a spanning tree with minimum total weight.
• Decision Version: Given an edge-weighted graph G and a positive integer k, does G contains a spanning tree with total weight < k.
Nondeterministic Algorithm
• Guess a spanning tree T.• Check whether the total weight of T < k.
This is not clear!
How to guess a spanning tree?
• Guess n-1 edges where n is the number of vertices of G.
• Check whether those n-1 edges form a connected spanning subgraph, i.e., there is a path between every pair of vertices.
Co-decision version of MST
• Given an edge-weighted graph G and a positive integer k, does G contain no spanning tree with total weight < k?
Algorithm
• Computer a minimum spanning tree. • Check whether its weight > k. If yes, the
algorithm halts.
co-NP
• co-NP = {A | Σ* - A ε NP}
NP ∩ co-NP
So far, no natural problem has been found in NP ∩ co-NP, but not in P.
Linear Programming
• Decision version: Given a system of linear inequality, does the system have a solution?
• It was first proved in NP ∩ co-NP and later found in P (1979).
Primality Test
• Given a natural number n, is n a prime?
• It was first proved in NP ∩ co-NP and later found in P (2004).
Therefore
• A natural problem belonging to NP ∩ co-NP is a big sign for the problem belonging to P.
Proving a problem in NP
• In many cases, it is not hard.• In a few cases, it is not easy.
Integer Programming
• Decision version: Given A and b, does Ax > b contains an integer solution?
• The difficulty is that the domain of “guess” is too large.
Lecture 26 Polynomial-time many-one reduction
A < m B
• A set A in Σ* is said to be polynomial-time many-one reducible to B in Γ* if there exists a polynomial-time computable function f: Σ* → Γ* such that
x ε A iff f(x) ε B.
p
A = Hamiltonian cycle (HC)
• Given a graph G, does G contain a Hamiltonian cycle?
B = decision version of Traveling Salesman Problem (TSP)
• Given n cities and a distance table between these n cities, find a tour (starting from a city and come back to start point passing through each city exactly once) with minimum total length.
• Given n cities, a distance table and k > 0, does there exist a tour with total length <
k?
HC < m TSP
• From a given graph G, we need to construct (n cities, a distance table, k).
p
SAT < m 3-SAT
• SAT: Given a Boolean formula F, does F have a satisfied assignment?
• An assignment is satisfied if it makes F =1.
• 3-SAT: Given a 3-CNF F, does F have a satisfied assignment?
p
Property of < m
• A < m B and B < m C imply A < m C
• A < m B and B ε P imply A ε P
p
p p p
p
NP-complete
• A set A is NP-hard if for any B in NP, B < m A.• A set A is NP-complete if it is in NP and
NP-hard.• A decision problem is NP-complete if its
corresponding language is NP-complete. • An optimization problem is NP-hard if its
decision version is NP-hard.
p
Cook Theorem
SAT is NP-complete
3-SAT is NP-complete
HC is NP-complete
Vertex-Cover is NP-complete
Proof of Cook Theorem
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