If you consider an independent set versus a vertex cover,
what you have is the following.
If you have the smallest possible vertex cover, that means you have selected
a minimum number of vertices so that each edge is next to at least one of those vertices.
If you were to remove those vertices from a graph, then no edges would remain
because every edge is connected to at least one of those vertices of the vertex cover,
so what remains is always an independent set.
And since you've selected the smallest possible vertex cover,
so the smallest number of vertices you need to remove,
what remains must be the largest set or a largest set of independent vertices.
And now that discovery is, of course, really great news for Alice,
but it's also even better news for Carol whose smile gets a bit bigger,
and, of course, also Bob because what they have now found out is this:
clique and independent set are really closely connected.
So if Bob discovers a good--or if any of them discovers a good algorithm for clique,
then they have not only solved "clique" but also "independent set."
So if just one of those two problems is solvable in polynomial time
or, in other words, if just one of those two problems turns out to be tractable,
then the other problem will be tractable as well.
And now we have also connected vertex cover to independent set
because we've basically figured out that finding the largest possible independent set
for a graph is almost the same as looking for the smallest possible vertex cover.
This also means that if you can find the smallest possible vertex cover,
then you have found the largest possible independent set
which we already know, through transforming the graph, let's you find
the largest possible clique.
So, actually, we can also draw this connection here
because we already know about these two connections.
What they have discovered is something that is commonly known as a reduction.
And we'll get more deeply into that in the next unit.
What a reduction is is basically a transformation between two problems
so that if you find out that one of them is tractable,
then the other one is tractable as well.
So now the big question is, for clique, vertex cover, and independent set,
are those three problems tractable--in which case
Alice, Bob, and Carol would really keep smiling--
or will it turn out that all of these problems are intractable--in which case
Bob, Alice, and Carol would tend to be rather unhappy?
So which one is it going to be?
Are all three going to end up very happy, are all three going to end up very sad,
or is there maybe something in between?
You can find out how that story continues in our next unit
of our Introduction to Theoretical Computer Science.
Plus, in the next unit, I will also tell you how you could quickly win a million dollars.
