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 }

CLIQUE = {<G, k> : graph G has a clique with at least k vertices}.

**Note –**

Since CLIQUE is NP => some NDTM_{CLIQUE} accepts CLIQUE.

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

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

**Correctness:**

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

Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the CS Theory Course at a student-friendly price and become industry ready.

## Recommended Posts:

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.