Polytime Manyone reduction: Clique to E-TM

Prerequisite – Clique is NP
A Polynomial-time reduction is a method for solving one problem using another.
E-TM = {<M> : M is a TM and L(M) = \phi}
CLIQUE = {<G, k> : graph G has a clique with at least k vertices}.

Note –
Since CLIQUE is NP => some NDTMCLIQUE accepts CLIQUE.

Reduction(<G, k>)
    construct the following machine M
    M(x):
        1. Run NDTMCLIQUE on input <G, k>.
    2. If NDTMCLIQUE accepts; M rejects x.
    3. Else; M accepts x.
    return <M>

We convert the instance <G, k> \in CLIQUE to a TM <M> \in E-TM. And <G, k> \notin CLIQUE to a TM <M> \notin E-TM.

Correctness:

i. <G, k> \in CLIQUE => M rejects all input x => L(M)= \phi => <M> \in E-TM.
ii. <G, k> \notin CLIQUE => M accepts all input x => L(M)\neq\phi => <M> \notin E-TM.

Hence, reduction is correct.

Polytime –
The reduction involves describing the construction of a new Turing machine M for input <G, k>. We don’t run the machine on the input. Hence, the reduction is polytime.

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