Question: BQP and QMA are examples of complexity classes most commonly associated with what type of Turing machine?

Answer: quantum

<hl> Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. <hl> #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.

Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using <hl> quantum <hl> Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.

question: BQP and QMA are examples of complexity classes most commonly associated with what type of Turing machine?, context: Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.


Question: What is the expression used to represent a complexity class of counting problems?

Answer: #P

Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. <hl> #P is an important complexity class of counting problems (not decision problems). <hl> Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.

Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. <hl> #P <hl> is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.

question: What is the expression used to represent a complexity class of counting problems?, context: Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.


Question: IP and AM are most commonly defined by what type of proof system?

Answer: Interactive

Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). <hl> Classes like IP and AM are defined using Interactive proof systems. <hl> ALL is the class of all decision problems.

Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using <hl> Interactive <hl> proof systems. ALL is the class of all decision problems.

question: IP and AM are most commonly defined by what type of proof system?, context: Other important complexity classes include BPP, ZPP and RP, which are defined using probabilistic Turing machines; AC and NC, which are defined using Boolean circuits; and BQP and QMA, which are defined using quantum Turing machines. #P is an important complexity class of counting problems (not decision problems). Classes like IP and AM are defined using Interactive proof systems. ALL is the class of all decision problems.


Question: What is an example of a measurement within a complexity class that would create a bigger set of problems if the bounds were relaxed?

Answer: computation time

<hl> For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. <hl> In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) <hl> computation time <hl> indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

question: What is an example of a measurement within a complexity class that would create a bigger set of problems if the bounds were relaxed?, context: For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.


Question: In what expression can one expect to find DTIME(n)

Answer: DTIME(n2)

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. <hl> In particular, although DTIME(n) is contained in DTIME(n2) , it would be interesting to know if the inclusion is strict. <hl> For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in <hl> DTIME(n2) <hl>, it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

In particular, although DTIME(n) is contained in <hl> DTIME(n2) <hl> , it would be interesting to know if the inclusion is strict.

question: In what expression can one expect to find DTIME(n), context: For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.


Question: What theorems are responsible for determining questions of time and space requirements?

Answer: time and space hierarchy theorems

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by <hl> the time and space hierarchy theorems respectively. <hl> They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the <hl> time and space hierarchy theorems <hl> respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

question: What theorems are responsible for determining questions of time and space requirements?, context: For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.


Question: Resources are constrained by hierarchy theorems to produce what?

Answer: a proper hierarchy on the classes defined

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. <hl> They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. <hl> Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce <hl> a proper hierarchy on the classes defined <hl> by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.

question: Resources are constrained by hierarchy theorems to produce what?, context: For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.


Question: What kind of statement is made in the effort of establishing the time and space requirements needed to enhance the ultimate number of problems solved?

Answer: quantitative statements

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. <hl> Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved. <hl>

For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make <hl> quantitative statements <hl> about how much more additional time or space is needed in order to increase the number of problems that can be solved.

question: What kind of statement is made in the effort of establishing the time and space requirements needed to enhance the ultimate number of problems solved?, context: For the complexity classes defined in this way, it is desirable to prove that relaxing the requirements on (say) computation time indeed defines a bigger set of problems. In particular, although DTIME(n) is contained in DTIME(n2), it would be interesting to know if the inclusion is strict. For time and space requirements, the answer to such questions is given by the time and space hierarchy theorems respectively. They are called hierarchy theorems because they induce a proper hierarchy on the classes defined by constraining the respective resources. Thus there are pairs of complexity classes such that one is properly included in the other. Having deduced such proper set inclusions, we can proceed to make quantitative statements about how much more additional time or space is needed in order to increase the number of problems that can be solved.


Question: What is the foundation for separation results within complexity classes?

Answer: time and space hierarchy theorems

<hl> The time and space hierarchy theorems form the basis for most separation results of complexity classes. <hl> For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in PSPACE.

The <hl> time and space hierarchy theorems <hl> form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in PSPACE.

question: What is the foundation for separation results within complexity classes?, context: The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in PSPACE.


Question: What is responsible for constraining P according to the time hierarchy theorem?

Answer: EXPTIME

The time and space hierarchy theorems form the basis for most separation results of complexity classes. <hl> For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME , and the space hierarchy theorem tells us that L is strictly contained in PSPACE. <hl>

The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem tells us that P is strictly contained in <hl> EXPTIME <hl>, and the space hierarchy theorem tells us that L is strictly contained in PSPACE.

For instance, the time hierarchy theorem tells us that P is strictly contained in <hl> EXPTIME <hl> , and the space hierarchy theorem tells us that L is strictly contained in PSPACE.

question: What is responsible for constraining P according to the time hierarchy theorem?, context: The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in PSPACE.


Question: Within what variable is L constrained according to the space hierarchy theorem?

Answer: PSPACE

The time and space hierarchy theorems form the basis for most separation results of complexity classes. <hl> For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in PSPACE . <hl>

The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in <hl> PSPACE <hl>.

For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in <hl> PSPACE <hl> .

question: Within what variable is L constrained according to the space hierarchy theorem?, context: The time and space hierarchy theorems form the basis for most separation results of complexity classes. For instance, the time hierarchy theorem tells us that P is strictly contained in EXPTIME, and the space hierarchy theorem tells us that L is strictly contained in PSPACE.


Question: What concept is frequently used to define complexity classes?

Answer: reduction

<hl> Many complexity classes are defined using the concept of a reduction . <hl> A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

Many complexity classes are defined using the concept of a <hl> reduction <hl>. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

Many complexity classes are defined using the concept of a <hl> reduction <hl> .

question: What concept is frequently used to define complexity classes?, context: Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.


Question: Reduction essentially takes one problem and converts into what?

Answer: another problem

Many complexity classes are defined using the concept of a reduction. <hl> A reduction is a transformation of one problem into another problem . <hl> It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into <hl> another problem <hl>. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

A reduction is a transformation of one problem into <hl> another problem <hl> .

question: Reduction essentially takes one problem and converts into what?, context: Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.


Question: According to reduction, if X and Y can be solved by the same algorithm then X performs what function in relationship to Y?

Answer: reduces

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that <hl> X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions. <hl>

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X <hl> reduces <hl> to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

question: According to reduction, if X and Y can be solved by the same algorithm then X performs what function in relationship to Y?, context: Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.


Question: What are two examples of different types of reduction?

Answer: Karp reductions and Levin reductions

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. <hl> For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions , and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions. <hl>

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, <hl> Karp reductions and Levin reductions <hl>, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, <hl> Karp reductions and Levin reductions <hl> , and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.

question: What are two examples of different types of reduction?, context: Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.


Question: Polynomial time reductions are an example of what?

Answer: the bound on the complexity of reductions

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. <hl> For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions , such as polynomial-time reductions or log-space reductions. <hl>

Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and <hl> the bound on the complexity of reductions <hl>, such as polynomial-time reductions or log-space reductions.

For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and <hl> the bound on the complexity of reductions <hl> , such as polynomial-time reductions or log-space reductions.

question: Polynomial time reductions are an example of what?, context: Many complexity classes are defined using the concept of a reduction. A reduction is a transformation of one problem into another problem. It captures the informal notion of a problem being at least as difficult as another problem. For instance, if a problem X can be solved using an algorithm for Y, X is no more difficult than Y, and we say that X reduces to Y. There are many different types of reductions, based on the method of reduction, such as Cook reductions, Karp reductions and Levin reductions, and the bound on the complexity of reductions, such as polynomial-time reductions or log-space reductions.


Question: What is the most frequently employed type of reduction?

Answer: polynomial-time reduction

<hl> The most commonly used reduction is a polynomial-time reduction . <hl> This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

The most commonly used reduction is a <hl> polynomial-time reduction <hl>. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

The most commonly used reduction is a <hl> polynomial-time reduction <hl> .

question: What is the most frequently employed type of reduction?, context: The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.


Question: What equates to a squared integer according to polynomial time reduction?

Answer: multiplying two integers

The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. <hl> For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers . <hl> This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of <hl> multiplying two integers <hl>. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

For example, the problem of squaring an integer can be reduced to the problem of <hl> multiplying two integers <hl> .

question: What equates to a squared integer according to polynomial time reduction?, context: The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.


Question: What measurement of time is used in polynomial time reduction?

Answer: polynomial time

The most commonly used reduction is a polynomial-time reduction. <hl> This means that the reduction process takes polynomial time . <hl> For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes <hl> polynomial time <hl>. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

This means that the reduction process takes <hl> polynomial time <hl> .

question: What measurement of time is used in polynomial time reduction?, context: The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.


Question: What would need to remain constant in a multiplication algorithm to produce the same outcome whether multiplying or squaring two integers?

Answer: input

The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. <hl> Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. <hl> Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same <hl> input <hl> to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

question: What would need to remain constant in a multiplication algorithm to produce the same outcome whether multiplying or squaring two integers?, context: The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.


Question: According to polynomial time reduction squaring can ultimately be logically reduced to what?

Answer: multiplication

The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. <hl> Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. <hl> Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the <hl> multiplication <hl> algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.

question: According to polynomial time reduction squaring can ultimately be logically reduced to what?, context: The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication.


Question: The complexity of problems often depends on what?

Answer: the type of reduction being used

This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. <hl> Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used . <hl> For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.

This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on <hl> the type of reduction being used <hl>. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.

Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on <hl> the type of reduction being used <hl> .

question: The complexity of problems often depends on what?, context: This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.


Question: What would create a conflict between a problem X and problem C within the context of reduction?

Answer: if every problem in C can be reduced to X

This motivates the concept of a problem being hard for a complexity class. <hl> A problem X is hard for a class of problems C if every problem in C can be reduced to X . <hl> Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.

This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C <hl> if every problem in C can be reduced to X <hl>. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.

A problem X is hard for a class of problems C <hl> if every problem in C can be reduced to X <hl> .

question: What would create a conflict between a problem X and problem C within the context of reduction?, context: This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.


Question: An algorithm for X which reduces to C would us to do what?

Answer: solve any problem in C

This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. <hl> Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C . <hl> Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.

This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to <hl> solve any problem in C <hl>. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.

Thus no problem in C is harder than X, since an algorithm for X allows us to <hl> solve any problem in C <hl> .

question: An algorithm for X which reduces to C would us to do what?, context: This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.


Question: A problem set that that is hard for the expression NP can also be stated how?

Answer: NP-hard

This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. <hl> In particular, the set of problems that are hard for NP is the set of NP-hard problems. <hl>

This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of <hl> NP-hard <hl> problems.

question: A problem set that that is hard for the expression NP can also be stated how?, context: This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems.


Question: The hardest problems in NP can be analogously written as what class of problems?

Answer: NP-complete

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) <hl> Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem <hl> P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of <hl> NP-complete <hl> problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.

question: The hardest problems in NP can be analogously written as what class of problems?, context: If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.


Question: NP complete problems contain the lowest likelihood of being located in what problem class?

Answer: NP

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) <hl> Thus the class of NP -complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem <hl> P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of <hl> NP <hl>-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.

Thus the class of <hl> NP <hl> -complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem

question: NP complete problems contain the lowest likelihood of being located in what problem class?, context: If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.


Question: If P = NP is unsolved, and reduction is applied to a known NP-complete problem vis a vis Π2 to Π1, what conclusion can be drawn for Π1?

Answer: there is no known polynomial-time solution

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = <hl> NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. <hl> This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that <hl> there is no known polynomial-time solution <hl> for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.

question: If P = NP is unsolved, and reduction is applied to a known NP-complete problem vis a vis Π2 to Π1, what conclusion can be drawn for Π1?, context: If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.


Question: If polynomial time can be utilized within an NP-complete problem, what does the imply P is equal to?

Answer: NP

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) <hl> Thus the class of NP -complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem <hl> P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.

If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of <hl> NP <hl>-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.

Thus the class of <hl> NP <hl> -complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem

question: If polynomial time can be utilized within an NP-complete problem, what does the imply P is equal to?, context: If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP.


Question: What complexity class is characterized by a computational tasks and efficient algorithms?

Answer: P

<hl> The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. <hl> This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

The complexity class <hl> P <hl> is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

question: What complexity class is characterized by a computational tasks and efficient algorithms?, context: The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.


Question: What hypothesis is associated with the complexity class of P viewed as a mathematical abstraction with efficient algorithmic functionality?

Answer: Cobham–Edmonds thesis

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. <hl> This hypothesis is called the Cobham–Edmonds thesis . <hl> The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the <hl> Cobham–Edmonds thesis <hl>. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

This hypothesis is called the <hl> Cobham–Edmonds thesis <hl> .

question: What hypothesis is associated with the complexity class of P viewed as a mathematical abstraction with efficient algorithmic functionality?, context: The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.


Question: What complexity class is commonly characterized by unknown algorithms to enhance solvability?

Answer: NP

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. <hl> The complexity class NP , on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. <hl> Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class <hl> NP <hl>, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

The complexity class <hl> NP <hl> , on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem.

question: What complexity class is commonly characterized by unknown algorithms to enhance solvability?, context: The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.


Question: What is an example of a problem that rests within the NP complexity class?

Answer: Boolean satisfiability problem

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. <hl> The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem , the Hamiltonian path problem and the vertex cover problem. <hl> Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the <hl> Boolean satisfiability problem <hl>, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the <hl> Boolean satisfiability problem <hl> , the Hamiltonian path problem and the vertex cover problem.

question: What is an example of a problem that rests within the NP complexity class?, context: The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.


Question: In what theoretical machine is it confirmed that a problem in P belies membership in the NP class?

Answer: Turing machines

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. <hl> Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP. <hl>

The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic <hl> Turing machines <hl> are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.

question: In what theoretical machine is it confirmed that a problem in P belies membership in the NP class?, context: The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP.


Question: If P is ultimately proven to be equal tot NP, what effect would this have on the efficiency of problems?

Answer: more efficient solutions

The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. <hl> If the answer is yes, many important problems can be shown to have more efficient solutions . <hl> These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.

The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have <hl> more efficient solutions <hl>. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.

If the answer is yes, many important problems can be shown to have <hl> more efficient solutions <hl> .

question: If P is ultimately proven to be equal tot NP, what effect would this have on the efficiency of problems?, context: The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.


Question: What is a particular problem in biology that would benefit from determining that P = NP?

Answer: protein structure prediction

The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. <hl> These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. <hl> The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.

The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, <hl> protein structure prediction <hl> in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.

question: What is a particular problem in biology that would benefit from determining that P = NP?, context: The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.


Question: What is the prize offered for finding a solution to P=NP?

Answer: $1,000,000

The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. <hl> There is a US $1,000,000 prize for resolving the problem. <hl>

The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US<hl> $1,000,000 <hl> prize for resolving the problem.

There is a US <hl> $1,000,000 <hl> prize for resolving the problem.

question: What is the prize offered for finding a solution to P=NP?, context: The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem.


Question: Who demonstrated that P= NP implies problems not present in P or NP-complete?

Answer: Ladner

<hl> It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. <hl> Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.

It was shown by <hl> Ladner <hl> that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.

question: Who demonstrated that P= NP implies problems not present in P or NP-complete?, context: It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.


Question: What is the name for a problem that meets Ladner's assertion?

Answer: NP-intermediate problems

It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. <hl> Such problems are called NP-intermediate problems . <hl> The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.

It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called <hl> NP-intermediate problems <hl>. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.

Such problems are called <hl> NP-intermediate problems <hl> .

question: What is the name for a problem that meets Ladner's assertion?, context: It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.


Question: What is an example of an NP-intermediate problem not known to exist in P or NP-complete?

Answer: graph isomorphism problem

It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. <hl> The graph isomorphism problem , the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. <hl> They are some of the very few NP problems not known to be in P or to be NP-complete.

It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The <hl> graph isomorphism problem <hl>, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.

The <hl> graph isomorphism problem <hl> , the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate.

question: What is an example of an NP-intermediate problem not known to exist in P or NP-complete?, context: It was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are examples of problems believed to be NP-intermediate. They are some of the very few NP problems not known to be in P or to be NP-complete.


Question: What is the problem attributed to defining if two finite graphs are isomorphic?

Answer: The graph isomorphism problem

<hl> The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. <hl> An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.

<hl> The graph isomorphism problem <hl> is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.

question: What is the problem attributed to defining if two finite graphs are isomorphic?, context: The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.


Question: What class is most commonly not ascribed to the graph isomorphism problem in spite of definitive determination?

Answer: NP-complete

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. <hl> An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete , or NP-intermediate. <hl> The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, <hl> NP-complete <hl>, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.

An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, <hl> NP-complete <hl> , or NP-intermediate.

question: What class is most commonly not ascribed to the graph isomorphism problem in spite of definitive determination?, context: The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.


Question: What finite hierarchy implies that the graph isomorphism problem is NP-complete?

Answer: polynomial time hierarchy

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. <hl> If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. <hl> Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the <hl> polynomial time hierarchy <hl> collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.

question: What finite hierarchy implies that the graph isomorphism problem is NP-complete?, context: The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.


Question: To what level would the polynomial time hierarchy collapse if graph isomorphism is NP-complete?

Answer: second level

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. <hl> If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level . <hl> Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its <hl> second level <hl>. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.

If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its <hl> second level <hl> .

question: To what level would the polynomial time hierarchy collapse if graph isomorphism is NP-complete?, context: The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.


Question: Who are commonly associated with the algorithm typically considered the most effective with respect to finite polynomial hierarchy and graph isomorphism?

Answer: Laszlo Babai and Eugene Luks

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. <hl> The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices. <hl>

The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to <hl> Laszlo Babai and Eugene Luks <hl> has run time 2O(√(n log(n))) for graphs with n vertices.

question: Who are commonly associated with the algorithm typically considered the most effective with respect to finite polynomial hierarchy and graph isomorphism?, context: The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P, NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least not NP-complete. If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its second level. Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believed that graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and Eugene Luks has run time 2O(√(n log(n))) for graphs with n vertices.