Concepts inInteractive proofs and the hardness of approximating cliques

Interactive proof system

In computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two parties. The parties, the verifier and the prover, interact by exchanging messages in order to ascertain whether a given string belongs to a language or not. The prover is all-powerful and possesses unlimited computational resources, but cannot be trusted, while the verifier has bounded computation power.
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NP (complexity)

In computational complexity theory, NP is one of the most fundamental complexity classes. The abbreviation NP refers to "nondeterministic polynomial time. " Intuitively, NP is the set of all decision problems for which the instances where the answer is "yes" have efficiently verifiable proofs of the fact that the answer is indeed "yes. " More precisely, these proofs have to be verifiable in polynomial time by a deterministic Turing machine.
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Hardness of approximation

In computer science, hardness of approximation is a field that studies the algorithmic complexity of finding near-optimal solutions to optimization problems. It complements the study of approximation algorithms by proving, for certain problems, a limit on the factors with which their solution can be efficiently approximated.
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Correctness (computer science)

In theoretical computer science, correctness of an algorithm is asserted when it is said that the algorithm is correct with respect to a specification. Functional correctness refers to the input-output behaviour of the algorithm (i.e. , for each input it produces the correct output). A distinction is made between total correctness, which additionally requires that the algorithm terminates, and partial correctness, which simply requires that if an answer is returned it will be correct.
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Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges. Typically, a graph is depicted in diagrammatic form as a set of dots for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.
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Bit

A bit (a contraction of binary digit) is the basic capacity of information in computing and telecommunications; a bit represents either 1 or 0 (one or zero) only. The representation may be implemented, in a variety of systems, by means of a two state device. In computing, a bit can be defined as a variable or computed quantity that can have only two possible values. These two values are often interpreted as binary digits and are usually denoted by the numerical digits 0 and 1.
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Function (mathematics)

In mathematics, a function is a relation between a set of inputs and a set of potential outputs with the property that each input is related to exactly one output. An example of such a relation is defined by the rule f(x) = x, which relates an input x to its square, which are both real numbers. The output of the function f corresponding to an input x is denoted by f(x) (read "f of x"). If the input is –3, then the output is 9, and we may write f(–3) = 9.
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