NP-Complete
Problems
NP and P
• What is NP?
• NP is the set of all decision problems (question with yesor-no answer) for which the 'yes'-answers can be verified
in polynomial time (O(n^k) where n is the problem size,
and k is a constant) by a deterministic Turing machine.
Polynomial time is sometimes used as the definition of
fast or quickly.
• What is P?
• P is the set of all decision problems which can be solved
in polynomial time by a deterministic Turing machine.
Since it can solve in polynomial time, it can also be
verified in polynomial time. Therefore P is a subset of NP.
•
Class
of
“P”
Problems
Class P consists of (decision) problems that are
solvable in polynomial time
• Polynomial-time algorithms
o Worst-case running time is O(nk), for some constant k
• Examples of polynomial time:
o O(n2), O(n3), O(1), O(n lg n)
• Examples of non-polynomial time:
o O(2n), O(nn), O(n!)
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•
Class
of
“NP”
Problems
Class NP consists of problems that could be solved
by NP algorithms
o i.e., verifiable in polynomial time
• If we were given a “certificate” of a solution, we
could verify that the certificate is correct in time
polynomial to the size of the input
• Warning: NP does not mean “non-polynomial”
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NP-Complete
• What is NP-Complete?
• A problem x that is in NP is also in NP-Complete if
and only if every other problem in NP can be
quickly (ie. in polynomial time) transformed into x. In
other words:
• x is in NP, and
• Every problem in NP is reducible to x
• So what makes NP-Complete so interesting is that if
any one of the NP-Complete problems was to be
solved quickly then all NP problems can be solved
quickly
NP-Hard
• What is NP-Hard?
• NP-Hard are problems that are at least as hard as
the hardest problems in NP. Note that NP-Complete
problems are also NP-hard. However not all NP-hard
problems are NP (or even a decision problem),
despite having 'NP' as a prefix. That is the NP in NPhard does not mean 'non-deterministic polynomial
time'. Yes this is confusing but its usage is
entrenched and unlikely to change.
Tractable/Intractable
Problems
• Problems in P are also called tractable
• Problems not in P are intractable or unsolvable
o Can be solved in reasonable time only for small inputs
o Or, can not be solved at all
• Are non-polynomial algorithms always worst than
polynomial algorithms?
- n1,000,000 is technically tractable, but really impossible - nlog log log n is
technically intractable, but easy
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Reductions
• Reduction is a way of saying that one problem is
“easier” than another.
• We say that problem A is easier than problem B,
(i.e., we write “A  B”)
if we can solve A using the algorithm that solves B.
• Idea: transform the inputs of A to inputs of B

f

yes
Problem B
no
Problem A
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yes
no
Polynomial Reductions
Given two problems A, B, we say that A is
•
polynomially reducible to B (A p B) if:
1.
There exists a function f that converts the input of A to inputs of B in
polynomial time
2.
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A(i) = YES  B(f(i)) = YES
NP-Complete Problems
Traveling Salesman
5-Clique
Hamiltonian Path
Map Coloring
Vertex Cover (VC)
• Given a graph and an integer k, is there a
collection of k vertices such that each edge is
connected to one of the vertices in the collection?