HALF CLIQUE - NP Complete Problem

Question

Let me start off by noting this is a homework problem, please provide only advice and related observations, NO DIRECT ANSWERS please. With that said, here is the problem I am looking at:

Let HALF-CLIQUE = { $\langle G \rangle$ | $G$ is an undirected graph having a complete subgraph with at least $n/2$ nodes, where n is the number of nodes in $G$ }. Show that HALF-CLIQUE is NP-complete.

Also, I know the following:

• In terms of this problem a clique, is defined as an undirected subgraph of the input graph, wherein every two nodes are connected by an edge. A $k$-clique is a clique that contains $k$ nodes.
• According to our textbook, Michael Sipser's "Introduction to the Theory of Computation", pg 268, that the problem CLIQUE = {$\langle G,k\rangle$ | $G$ is an undirected graph with a $k$-clique} is in NP
• Furthermore, according to the same source (on pg 283) notes that CLIQUE is in NP-Complpete (thus also obviously in NP).

I think I have the kernel of an answer here, however I could use some indication of what is wrong with it or any related points that might be relevant to an answer. This is the general idea I have so far,

Ok, I'd first note that a certificate would simply be a HALF-QLIQUE of $\text{size} \geq n/2$. Now it appears that what I would need to do is to create a verifier that is a polynomial time reduction from CLIQUE (which we know is NP-Complete) to HALF-CLIQUE. My idea would be that this would be done by creating a Turing machine which runs the turing machine verifier in the book for CLIQUE with the additional constraint for HALF-CLIQUE.

This sounds correct to me, but I don't really trust myself yet in this subject. Once again, I would like to remind everyone this is a HOMEWORK PROBLEM so please try to avoid answering the question. Any guidance which falls short of this would be most welcome!

Answer 1

Judging by your description and comments, you might be helped the best by an exact description on how reductions can be used to prove NP-completeness:

A problem is NP-complete iff it is in NP and it is NP-hard. This means that any proof of NP-completeness has two parts: a proof that the problem lies in NP and a proof that it is NP-hard.

For the first part, you have to show that YES-instances can be verified in polynomial time using some suitable certificate. Alternatively, you can show the problem can be solved in polynomial time by a non-deterministic Turing machine, but this is not usually done as mistakes are easily made.

In your case, this comes down to proving that for every graph with a $n/2$-clique, you can find some proof that there is indeed such a clique, such that, armed with such a proof, you can check in polynomial time that there is indeed such a clique.

For the second part, you have to show that the problem is NP-hard. This is in nearly all cases shown by proving that your problem is at least as hard as some other NP-hard problem. If HALF-CLIQUE is as least as hard as CLIQUE, it must also be NP-hard.

You do this by proving a reduction FROM CLIQUE, TO HALF-CLIQUE. You 'reduce' the problem, making it 'easier'. You say "Solving CLIQUE is hard, but I have proven that you only need to solve HALF-CLIQUE to solve CLIQUE". (many people, even experts, occasionally say this the wrong way around :))

There are different types of reductions: the reduction that is most commonly used is the one where you map instances of in this case CLIQUE to instances of HALF-CLIQUE whose size is at most polynomially larger, in polynomial time. This means that if we can solve HALF-CLIQUE, then we can also solve CLIQUE by chaining the algorithm and the reduction.

In other words, we must show that we can solve CLIQUE if we can solve HALF-CLIQUE. We do this by showing that for every instance for CLIQUE, we can design an instance of HALF-CLIQUE such that the instance of CLIQUE is a 'yes' instance iff the instance of HALF-CLIQUE is a 'yes' instance.

The proof therefore starts like this: given a graph $G=(V, E)$, I can create some graph $H=(V', E')$ such that $G$ contains a $k$-clique iff $H$ contains a $n/2$-clique. I'll leave this part to you (this is the part that requires creativity, the part that is about the specific problem at hand).

Answer 2

You seem a little lost. You want to show that HALF-CLIQUE $\ge$ CLIQUE, which means that you are looking for a poly-time algorithm that takes CLIQUE instances as the input and outputs HALF-CLIQUE instances with the property that "yes" inputs map to "yes"es and "no" inputs map to "no"s.

So, basically, the task is to take a graph $G$ and a number $k$ and output a new graph $H$ (and no number) so that $H$ has a clique on at least half of its vertices whenever $G$ has a clique of size $k$.

The spoiler below contains a hint on how to perform this reduction:

Try to make $H$ by attaching (in some way) an appropriately sized clique to $G$, plus some number of vertices not connected to anything.

Answer 3

You can reduce from the vertex cover problem. If the complement graph of the given graph has a vertex cover less than n/2 nodes then this graph will have a clique of more than n/2 nodes that is it will be a half clique. Just state that it is difficult to solve the Vertex Cover problem so is this.