CSCI 4325 / 6339 Theory of Computation

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Transcript of CSCI 4325 / 6339 Theory of Computation
CSCI 4325 / 6339Theory of Computation
Zhixiang Chen
Chapter 6 Computational Complexity
The New World
P
NP
NPcomplete
Polynomially Bounded TM’s
Definition. A TM M=(K,∑,δ,s,H) is said to be polynomially bounded if there is polynomial p(n) such that the following is true: For any input x, the computation of M on x ends in
no more p(x)+1 steps.
Polynomially Decidable Languages Definition. A language is called polynomially
decidable is there is a polynomially bounded TM that decides it.
The Class P
Definition. The class P is the set of all polynomially decidable languages.
Examples. The set of even integers The set of connected graphs Other examples?
Quantitative Refinement of the ChurchTuring Thesis Any problem which is intuitively feasibly
solvable is solvable by a polynomially bounded TM.
Any problem which is solvable by a polynomially bounded TM is intuitively feasibly solvable.
Theorem
Theorem 6.1.1. The class P is closed under complement.
Proof. (Prove in class.)
Something not in P
Definition. Define E={encode(M)encode(w):
M accepts w in but steps}
Theorem. Proof. By contradiction. Consider a special case of E:
M accepts encode(M) but
steps}
Prove
12 w
2 w
.PE
:)({1 MencodeE 2 w
12 w
.1 PE
New Notations
For simplicity We use “M” to denote the encode(M) of a TM M We also use “w” to denote the encode(w) of a
string w.
Some Coments
Comments on polynomial time bounded decidability Practically feasible? It’s based on worstcase performance analysis How about averagecase performance analysis?
Problems vs. Languages
Given any TM M and any input w, will M halts on w? This is a problem.
Define This is a language.
The fact The problem and the language are equivalent!
}:"""{" wonhaltsMwMH
Problems vs. Languages
Given any string w, will w have property P? This is a problem.
Define This is a language.
The fact again The problem and the language are equivalent!
Conclusion: A problem is a language. A language is a problem.
}:{ PpropertyhaswwH
Eulerian Graphs
Euler Cycle Problem: Given a graph G, is there a closed path in G that uses each edge exactly once?
A graph that has a Euler cycle is called Eulerian.
Ex’s
Eulerian Graphs
Euler Cycle Problem is in P. That is,
is in P.
Proof. By Euler Theorem: Euler Theorem. A graph G is Eulerian iff the following two
conditions are true: For any pair of nodes u, v in G, there is a path from u to v. All nodes have equal numbers of incoming and outgoing
edges
}:)({ EulerianisGGL
Hamiltonian Graphs
Hamilton Cycle Problem: Given a graph G, is there a cycle that passes through each node of G exactly once?
A graph with a Hamilton cycle is called Hamiltonian
Is Hamilton Cycle Problem in P? Do you know the answer? Do you have an algorithm for solving Hamilton
Cycle Problem? What is the complexity of your algorithm?
Yes/No Problems
A yes/no problem is a problem with a yes or no answer.
Euler Cycle Problem and Hamilton Cycle Problem are yes/no problems.
Optimization Problems
Traveling Salesman Problem. Given a weighted graph G, find a shortest path touring all nodes in a graph.
Translate Optimization Problem to Yes/No Problem: Given an integer and an distance matrix and
an integer find a Hamiltonian cycle with distance
The above optimization problem and the yes/no problem are “equivalent”.
In general, any optimization problem can translated to an “equivalent” yes/no problem.
nn
.B
,2n ,ijd
,0B
Independent Set Problem
Given an undirected graph G=(V,E) and an integer
is there a subset C of V with
such that for all there is no edge between and
Ex’s?
KC 
Vvu ji ,
,2K
iu ?jv
Clique Problem
Given an undirected graph G=(V,E) and an integer
is there a subset C of V with
such that for all there is an edge between and
Ex’s?
KC Cvu ji ,
,2K
iu ?jv
Node Cover Problem
Given an undirected graph G=(V,E) and an integer
is there a subset C of V with
such that C covers all edges of G?
Note: C covers an edge if it contains at least on endpoint of the edge.
Ex’s?
BC ,2B
Integer Partition Problem
Given a set of n nonnegative integers
represented in binary, is there a subset
such that
Ex’s?
},,2,1{ nP
?
Pi
iPi
i aa
naaa ,,, 21
Solution to Integer Partition Problem Compute If H is not an integer then stop and say no Otherwise compute B(i) for
b is the sum of some subset of }
Algorithm for computing B(i): B(0)={0} For do
For do
If then add to B(i)
:{)( HbiB },,{ 1 iaa
n
iiaH
12
1
)1()( iBiB
ni ,,2,1
Haaaj iii ,,2,1, )1( iBaj i
j
Example
Work on the following set of integers to determine the answer to the partition problem.
H = 151 B(0) = {0} B(1) = {0, 38} B(2) = {0, 17, 38, 55} B(3) = {0, 17, 38, 52, 55, 69, 90, 107} B(4) = {0, 17, 38, 52, 55, 61, 69, 78, 90, 107, 113, 116, 130, 151} … B(7) = {…, 151} Answer: yes.
.25,88,21
,61,52,17,38
765
4321
aaa
aaaa
Complexity of the Algorithm for Computing B(i) The complexity if O(nH)
Outer loop runs n times Inner loop runs H times
So, it the Partition Problem in P? Just from O(nH), it seems the problem in P. What is wrong???????
Boolean Satisfiability Problem Definition.
Let be a finite set of Boolean variables, and let be the set of negations of variables in X.
Call elements in as literals
Negations of variables are called negative literals.
},,,{ 21 nxxxX
},,,{ 21 nxxxX
XX
Boolean Satisfiability Problem Definition.
Clause: A clause C is a nonempty set of literals, i.e.
Boolean formula in CNF: A Boolean CNF is a set of clauses
.XXC
Examples
Let
Clauses:
CNF:
},,,{},,,{ 321321 xxxXxxxX
},,{},,,{ 321321 xxxxxx
}},,{},,,{},,{{ 32132121 xxxxxxxx
Boolean Satisfiability Problem Truth Assignment:
a map from X to {0,1} 0 means false 1 means true
A clause is true under an assignment if that assignment make at least of its literals true.
A CNF is true under an assignment if that assignment makes all clauses true.
A CNF is satisfiable, if there is an assignment making it true.
Ex’s?
.XXC
Boolean Satisfiability Problem The SAT Problem: Given a Boolean CNF F,
is F satisfiable? The 2SAT Problem: Given a Boolean CNF F
such that each clause in F has at most two literals, is F satisfiable?
The 3SAT Problem: Given a Boolean CNF F such that each clause in F has at most three literals, is F satisfiable?
2SAT Problem
Theorem. 2SAT is in P.
2SAT Problem Algorithm solving 2SAT
Phase 1. Purge Process Repeat
Scan clause in the input CNF F to find a single literal x Set x to be true. And remove all clauses with x from F, and
remove negative x from these clause containing negative x Fail if an empty clause is generated, and stop and say no.
Until no more single clause in F. Phase 2. Try Two Values Process
Repeat Choose a literal x in the remained F
Set x to be true, and do “Purge Process” Set x to be false and do “purge Process”
If both processes fail, then stop and say no. Until no more literal left.
Phase 3. Final Checking Say true, if no fail occurs in Phase 2.
2SAT Problem
The complexity of previous algorithm Let n be the number of distinct variables in F Let m be the number of clauses in F Phase 1 Time
Phase 2 time
Total time
Conclusion. 2Sat is in P.
)( 2mnO
)( 2mnO
)(nmO
Nondeterministic Ploynomially Bounded TM’s Definition. A NTM M=(K,∑,Δ,s,H) is said to
be polynomially bounded is there is polynomial p(n) such that the following is true: For any input x, all possible computations of M on
x ends in no more p(x)+1 steps. In other words, the depth of the computation tree
of M on x is no more than p(x)+1.
The NP Class
Definition. The class NP is the set of all languages that are decided by ploynomially bounded NTM’s
PolyBounded Computation Trees )_,( s
),( vauh
………………Tree depth p(w)
Problems in NP
Problems in NP: SAT Hamilton Cycle Problem TSP Independent Set Problem Clique Problem Covering Problem Partition Problem
Proving a Problem in NP
Two steps: Nondeterministically guess an answer in polytime Deterministically verify the guessed answer is
indeed an answer in polytime
Show proof examples in class.
Exponentially Bounded Deterministic TM’s Definition. A TM M=(K,∑,δ,s,H) is said to be
exponentially bounded is there is polynomial p(n) such that the following is true: For any input x, the computation of M on x ends in no
more than steps.)(2 xp
The Class EXP
Definition. The class EXP is the set of all problems that are decidable by exponentiallybounded TM’s.
P, NP, vs. EXP
Theorem. If L is in NP, then L is in EXP. Proof. A polybounded computation tree can be
simulated an exponentially bounded computation.
Theorem. Proof. Consider the exponentially bounded version
of H.
.EXPP
The New World
P
NP
EXP
Open Questions
EXPNP
NPP
?
?