1. Theory of Computability
The class NP
Section 7.3

2. and target node are fixed.
The HAMPATH problem
A Hamiltonian path in a directed graph G is a directed path that goes through each
node exactly once. We consider a special case of this problem where the start node
and target node are fixed.
HAMPATH = {<G,s,t> | G is a directed graph with a Hamiltonian path from s to t} 1 5 3
 PATH? s t 4 2 6 1 5 3
 HAMPATH? s t 4 2 6

3. Polynomial verifiability
Does this graph have a
Hamiltonian path? s t

4. The COMPOSITES problem
COMPOSITES = {x | x = pq for some integers p, q > 1}
15  COMPOSITES ?
17  COMPOSITES ?
77  COMPOSITES ?
10403  COMPOSITES ?
COMPOSITES is very easily seen to be verifiable in polynomial time.
But only a few years ago it was proven that it is also decidable in polynomial time (2002, N. Agrawal, N. Kayal and N. Saxena, claimed a deterministic polynomial-time algorithm).
The same question, however, remains open for HAMPATH --- one of the seven $1,000,000 questions!

5. Definition of a verifier
Definition: A verifier for a language A is an algorithm V, where
A = {w | V accepts <w, c> for some string c}.
We measure the time of a verifier only in terms of the length of w, so a polynomial
time verifier runs in polynomial time in the length of w. A language A is
polynomially verifiable if it has a polynomial time verifier. The above string c, used
as additional information to verify that w  A, is called a certificate, or proof, of
membership in A.
Observe that, for polynomial verifiers, the certificate has polynomial length (in the
length of w) because that is all the verifier can access in its time bound.
What is a certificate for <G,s,t>  HAMPATH?
What is a verifier for HAMPATH?
What is a certificate for x  COMPOSITES?
What is a verifier for COMPOSITES?

6. Definition of NP
Definition: NP is the class of languages that have polynomial time verifiers.
A very important class, because many natural and important problems are in it!
NP stand for “nondeterministic polynomial”. As it turns out, an
alternative characterization of NP is to say that this is the class of languages decidable
by polynomial-time nondeterministic Turing machines.
Here is such a machine for HAMPATH:
N1 = “On input <G,s,t>, where G is a directed graph with nodes s and t:
1. Write a list on m members p1,…,pm, where m is the number of nodes of G.
Each number in the list is nondeterministically selected to be between 1 and m.
2. Check for repetitions in the list. If any are found, reject.
3. Check whether s=p1 and t=pm. If either fail, reject.
4. For each i between 1 and m-1, check whether (pi,pi+1) is an edge of G. If any
are not, reject. Otherwise, all tests have been passed, so accept.”

7. NP in terms of nondeterministic Turing machines
Theorem: A language is in NP iff it is decided by some nondeterministic
polynomial time Turing machine.
Proof. The idea is to show how to convert a polynomial time verifier into a polynomial
time NTM, and vice versa. The NTM simulates the verifier by guessing the certificate.
The verifier simulates the NTM by using the accepting branch as the certificate.
() Assume A  NP. Let V be a verifier for A with VTIME(nk), which, by the
definition of NP, exists. Construct N as follows:
N = “On input w of length n:
1. Nondeterministically select a string c of length at most nk.
2. Run V on input <w, c>.
3. If V accepts, accept; otherwise reject.”
() Assume A is decided by a polynomial time NTM N. Construct a verifier V as
follows:
V = “On input <w ,c>, where w and c are strings:
Check whether c is (encodes) an accepting computation branch of N on input w.
If yes, accept; otherwise reject.”

8. Definition of NTIME(t(n))
Definition: Let t(n) be a function. NTIME(t(n)) is defined as
{L | L is a language decided by some O(t(n)) time NTM}.
Corollary:
NP = NTIME(n1)  NTIME(n2)  NTIME(n3)  NTIME(n4)  …
Just like P, the class NP is insensitive to the choice of the reasonable
underlying computational model because all such models are
polynomially equivalent. So, when describing and analyzing
nondeterministic polynomial algorithms, we can follow the preceding
(lazy man’s) conventions for deterministic polynomial time algorithms.

9. CLIQUE = {<G, k> | G is an undirected graph with a k-clique}
The CLIQUE problem
A k-clique in an undirected graph is a subgraph with
k nodes, wherein every two nodes are connected by
an edge. On the right we see a 5-clique
CLIQUE = {<G, k> | G is an undirected graph with a k-clique}
Theorem: CLIQUE  NP.
Idea: The clique is the certificate.
Proof. Here is a polynomial time verifier V for CLIQUE:
V = “On input <<G, k>, c>:
1. Test whether c is a set of k nodes in G.
2. Test whether G contains all edges connecting nodes in c.
3. If both pass, accept; otherwise reject.”
Alternative proof. Here is a polynomial time NTM N deciding CLIQUE:
N = “On input <G, k>:
1. Nondeterministically select a subset c of k nodes of G.
2. Test whether G contains all edges connecting nodes in c.
3. If yes, accept; otherwise reject.”

10. The SUBSET-SUM problem
SUBSET-SUM = {<S, t> | S is a multiset of integers and, for some
R  S, the sum of all elements of R equals t}
Theorem SUBSET-SUMNP.
Idea: The subset is the certificate.
Proof. Here is a polynomial time verifier V for SUBSET-SUM:
V = “On input <<S, t>,c>:
1. Test whether c is a collection of numbers that sum to t.
2. Test whether S contains all the numbers in c.
3. If both pass, accept; otherwise reject.”
Alternative proof. Here is a polynomial time NTM N deciding
SUBSET-SUM:
N = “On input <S, t>:
1. Nondeterministically select a subset c of S.
2. Test whether the elements of c sum up to t.
3. If yes, accept; otherwise reject.”

11. P vs. NP vs. coNP vs. EXPTIME
Definitions
coNP = {L | L is the complement of some language in NP}
EXPTIME = TIME(2n1)  TIME(2n2)  TIME(2n3)  …
This is what we know:
P NP
coNP EXPTIME
NP  coNP ≠ EXPTIME
P: membership can be decided quickly
NP: membership can be verified quickly
coNP: membership can be refuted quickly
EXPTIME: none of the above
This is what we do not know:
P = NP?
NP = coNP?
P = NP  coNP?