In the study of the computational complexity of problems, the first concern of both computer scientists and computing professionals is whether a given problem can be solved in polynomial time by some algorithm.

*P*** ****,**** ***NP*** ****, and**** ***NP-***Complete Problems**

In the study of the
computational complexity of problems, the first concern of both computer
scientists and computing professionals is whether a given problem can be solved
in polynomial time by some algorithm.

**DEFINITION
1 **We say that an algorithm solves a problem in polynomial time** **if its worst-case time
efficiency belongs to ** O(p(n))** where

There are several reasons for drawing the
intractability line in this way. First, the entries of Table 2.1 and their
discussion in Section 2.1 imply that we cannot solve arbitrary instances of
intractable problems in a reasonable amount of time unless such instances are
very small. Second, although there might be a huge difference between the
running times in ** O(p(n))** for polynomials of
drastically different degrees, there are very few useful polynomial-time
algorithms with the degree of a polynomial higher than three. In addition,
polynomials that bound running times of algorithms do not usually have
extremely large coefficients. Third, polynomial functions possess many
convenient properties; in particular, both the sum and composition of two
polynomials are always polynomials too. Fourth, the choice of this class has
led to a development of an extensive theory called

remains the same for all
principal models of computations and all reasonable input-encoding schemes for
the problem under consideration.

We just touch on some basic
notions and ideas of complexity theory in this section. If you are interested
in a more formal treatment of this theory, you will have no trouble finding a
wealth of textbooks devoted to the subject (e.g., [Sip05], [Aro09]).

*P ***and NP Problems**

Most problems discussed in
this book can be solved in polynomial time by some algorithm. They include
computing the product and the greatest common divisor of two integers, sorting
a list, searching for a key in a list or for a pattern in a text string,
checking connectivity and acyclicity of a graph, and finding a minimum spanning
tree and shortest paths in a weighted graph. (You are invited to add more
examples to this list.) Informally, we can think about problems that can be
solved in polynomial time as the set that computer science theoreticians call ** P .** A more formal definition includes in

**DEFINITION
2 **Class** P **is a class of decision
problems that can be solved in

The restriction of ** P** to decision problems can be justified by the
following reasons. First, it is sensible to exclude problems not solvable in
polynomial time because of their exponentially large output. Such problems do
arise naturally— e.g., generating subsets of a given set or all the
permutations of

It is natural to wonder
whether *every* decision problem can be
solved in polynomial time. The answer to this question turns out to be no. In
fact, some decision problems cannot be solved at all by any algorithm. Such
problems are called ** undecidable**, as opposed to

Here is a surprisingly short
proof of this remarkable fact. By way of contra-diction, assume that ** A** is an algorithm that solves the halting
problem. That is, for any program

We can consider program ** P** as an input to itself and use the output of
algorithm

This is a contradiction
because neither of the two outcomes for program ** Q** is possible, which completes the proof.

Are there decidable but
intractable problems? Yes, there are, but the number of *known* examples is surprisingly small, especially of those that
arise naturally rather than being constructed for the sake of a theoretical
argument.

There are many important
problems, however, for which no polynomial-time algorithm has been found, nor
has the impossibility of such an algorithm been proved. The classic monograph
by M. Garey and D. Johnson [Gar79] contains a list of several hundred such
problems from different areas of computer science, mathematics, and operations
research. Here is just a small sample of some of the best-known problems that
fall into this category:

** Hamiltonian circuit problem **Determine whether a given
graph has a

** Traveling salesman problem **Find the shortest tour
through

** Knapsack problem **Find the most valuable subset
of

** Partition problem **Given

** Integer linear programming
problem **Find the maximum (or minimum)

Some of these problems are
decision problems. Those that are not have decision-version counterparts (e.g.,
the ** m**-coloring problem for the
graph-coloring problem). What all these problems have in common is an
exponential (or worse) growth of choices, as a function of input size, from
which a solution needs to be found. Note, however, that some problems that also
fall under this umbrella can be solved in polynomial time. For example, the
Eulerian circuit problem—the problem of the existence of a cycle that traverses
all the edges of a given graph exactly once—can be solved in

Another common feature of a
vast majority of decision problems is the fact that although solving such
problems can be computationally difficult, checking whether a proposed solution
actually solves the problem is computationally easy, i.e., it can be done in
polynomial time. (We can think of such a proposed solution as being randomly
generated by somebody leaving us with the task of verifying its validity.) For
example, it is easy to check whether a proposed list of vertices is a
Hamiltonian circuit for a given graph with ** n** vertices. All we need to check is that the
list contains

**DEFINITION
3 **A** ***nondeterministic algorithm*** **is a two-stage procedure that** **takes as its input an
instance ** I** of a decision problem and
does the following.

Nondeterministic (“guessing”)
stage: An arbitrary string ** S** is generated that can be
thought of as a candidate solution to the given instance

Deterministic (“verification”) stage: A
deterministic algorithm takes both ** I** and

We say that a
nondeterministic algorithm solves a decision problem if and only if for every
yes instance of the problem it returns yes on some execu-tion. (In other words,
we require a nondeterministic algorithm to be capable of “guessing” a solution
at least once and to be able to verify its validity. And, of course, we do not
want it to ever output a yes answer on an instance for which the answer should
be no.) Finally, a nondeterministic algorithm is said to be ** nondeterministic
polynomial** if the time efficiency of its verification stage is
polynomial

Now we can define the class
of *NP* problems.

**DEFINITION
4 **Class** ***NP*** **is the class of decision
problems that can be solved by** **nondeterministic polynomial algorithms. This
class of problems is called ** nonde-terministic polynomial**.

Most decision problems are in
*NP*. First of all, this class includes
all the problems in ** P** :

** P **⊆

This is true because, if a
problem is in ** P** , we can use the
deterministic polynomial-time algorithm that solves it in the
verification-stage of a nondeterministic algo-rithm that simply ignores string

This leads to the most
important open question of theoretical computer sci-ence: Is ** P** a proper subset of

Note that ** P** =

*NP ***-Complete Problems**

Informally, an *NP-*complete problem is a problem in *NP* that is as difficult as any other
problem in this class because, by definition, any other problem in *NP* can be reduced to it in polynomial
time (shown symbolically in Figure 11.6).

Here are more formal
definitions of these concepts.

**DEFINITION
5 **A decision problem** D**

**
**** t **maps all yes instances of

**
**** t **is computable by a polynomial
time algorithm

This definition immediately
implies that if a problem *D*_{1} is polynomially reducible to
some problem *D*_{2} that can be solved in
polynomial time, then problem *D*_{1}** **can also be solved in polynomial time (why?).

**DEFINITION
6** A decision problem ** D** is said to be

**
**it belongs to class *NP*

**
**every problem in *NP* is
polynomially reducible to *D*

The fact that closely related
decision problems are polynomially reducible to each other is not very
surprising. For example, let us prove that the Hamiltonian circuit problem is
polynomially reducible to the decision version of the traveling

**11.3 ***P*** **,** ***NP*** **, and** ***NP-*Complete Problems

salesman problem. The latter
can be stated as the existence problem of a Hamil-tonian circuit not longer
than a given positive integer ** m** in a given complete graph
with positive integer weights. We can map a graph

Let ** G** be a yes instance of the Hamiltonian circuit
problem. Then

The notion of *NP-*completeness requires, however,
polynomial reducibility of *all *problems
in* NP, *both known and unknown, to the
problem in question. Given* *the
bewildering variety of decision problems, it is nothing short of amazing that
specific examples of *NP-*complete
problems have been actually found. Neverthe-less, this mathematical feat was
accomplished independently by Stephen Cook in the United States and Leonid
Levin in the former Soviet Union.^{2} In his 1971 paper, Cook [Coo71] showed that
the so-called ** CNF-satisfiability problem** is

The CNF-satisfiability
problem asks whether or not one can assign values *true* and *false *to
variables of a given boolean expression in its CNF form to make the entire* *expression *true*. (It is easy to see that this can be done for the above
formula: if *x*_{1}** **=

Since the Cook-Levin
discovery of the first known *NP-*complete
problems, computer scientists have found many hundreds, if not thousands, of
other exam-ples. In particular, the well-known problems (or their decision
versions) men-tioned above—Hamiltonian circuit, traveling salesman, partition,
bin packing, and graph coloring*—*are
all *NP-*complete. It is known,
however, that if ** P** =

For a while, the leading
candidate to be such an example was the problem of determining whether a given
integer is prime or composite. But in an im-portant theoretical breakthrough,
Professor Manindra Agrawal and his students Neeraj Kayal and Nitin Saxena of
the Indian Institute of Technology in Kanpur announced in 2002 a discovery of a
deterministic polynomial-time algorithm for primality testing [Agr04]. Their
algorithm does not solve, however, the related problem of factoring large
composite integers, which lies at the heart of the widely used encryption
method called the ** RSA algorithm** [Riv78].

Showing that a decision
problem is *NP-*complete can be done in
two steps. First, one needs to show that the problem in question is in *NP*; i.e., a randomly generated string
can be checked in polynomial time to determine whether or not it represents a
solution to the problem. Typically, this step is easy. The second step is to
show that every problem in *NP* is
reducible to the problem in question in polynomial time. Because of the
transitivity of polynomial reduction, this step can be done by showing that a
known *NP-*complete problem can be
transformed to the problem in question in polynomial time (see Figure 11.7).
Although such a transformation may need to be quite ingenious, it is
incomparably simpler than proving the existence of a transformation for every
problem in *NP*. For example, if we
already know that the Hamiltonian circuit problem is *NP-*complete, its polynomial reducibility to the decision traveling
salesman problem implies that the latter is also *NP-*complete (after an easy check that the decision traveling
salesman problem is in class *NP*).

The definition of *NP-*completeness immediately implies that
if there exists a deterministic polynomial-time algorithm for just one *NP-*complete problem, then every problem
in *NP* can be solved in polynomial
time by a deterministic algo-rithm, and hence ** P** =

for one *NP-*complete problem would mean that there is no qualitative
difference between the complexity of checking a proposed solution and finding
it in polyno-mial time for the vast majority of decision problems of all kinds.
Such implications make most computer scientists believe that ** P** =

?

Whatever the eventual answer
to the ** P** =

**Exercises 11.3**

**
**A game of chess can be posed as the following decision problem:
given a legal positioning of chess pieces and information about which side is
to move, determine whether that side can win. Is this decision problem
decidable?

**
**A certain problem can be solved by an algorithm whose running time
is in *O(n*^{log}^{2}* *^{n}** ). **Which of the following assertions is true?

**
**The problem is tractable.

**
**The problem is intractable.

**
**Impossible to tell.

**
**Give examples of the following graphs or explain why such examples
cannot exist.

**
**graph with a Hamiltonian circuit but without an Eulerian circuit

**
**graph with an Eulerian circuit but without a Hamiltonian circuit

**
**graph with both a Hamiltonian circuit and an Eulerian circuit

**
**graph with a cycle that includes all the vertices but with neither
a Hamil-tonian circuit nor an Eulerian circuit

**4. **For each of the following graphs, find its chromatic number.

**
**Design a polynomial-time algorithm for the graph 2-coloring
problem: deter-mine whether vertices of a given graph can be colored in no more
than two colors so that no two adjacent vertices are colored the same color.

**
**Consider the following brute-force algorithm for solving the
composite num-ber problem: Check successive integers from 2 to ** n/**2 as possible divisors of

**
**State the decision version for each of the following problems and
outline a polynomial-time algorithm that verifies whether or not a proposed
solution solves the problem. (You may assume that a proposed solution
represents a legitimate input to your verification algorithm.)

**
**knapsack problem**b. **bin packing
problem

**
**Show that the partition problem is polynomially reducible to the
decision version of the knapsack problem.

**
**Show that the following three problems are polynomially reducible
to each other.

Determine, for a given graph ** G** =

Determine, for a given graph ** G** =

Determine, for a given graph ** G** =

set of size ** k** for a graph

**
**Determine whether the following problem is *NP-*complete. Given several sequences of uppercase and lowercase
letters, is it possible to select a letter from each sequence without selecting
both the upper- and lowercase versions of any letter? For example, if the
sequences are Abc, BC, aB, and ac, it is possible to choose A from the first
sequence, B from the second and third, and c from the fourth. An example where
there is no way to make the required selections is given by the four sequences
AB, Ab, aB, and ab. [Kar86]

Which of the following
diagrams do not contradict the current state of our knowledge about the
complexity classes *P*, *NP*, and *NPC* (*NP-*complete
problems)?

**
**King Arthur expects 150 knights for an annual dinner at Camelot. Unfortu-nately,
some of the knights quarrel with each other, and Arthur knows who quarrels with
whom. Arthur wants to seat his guests around a table so that no two quarreling
knights sit next to each other.

**
**Which standard problem can be used to model King Arthur’s task?

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Introduction to the Design and Analysis of Algorithms : Limitations of Algorithm Power : P , NP , and NP-Complete Problems |

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