We're going to start off this unit with a small technical detail.
When we considered our three problems, vertex cover, independent set, and clique,
we always said that we were looking for the best possible solution.
So, for example, for the largest independent set or the smallest vertex cover.
This type of problem is known as an optimization problem
because we're trying to maximize or minimize some value.
In this unit, we're mostly going to work with a slightly different
version of these problems called decision problems.
So we're not going to ask, for example, what is the smallest possible vertex cover?
But rather we're going to ask, does a graph have a vertex cover that is smaller than some number of k?
And k is given to us in advance.
So the main difference between an optimization problem and a decision problem
is that for an optimization problem, we ask, find us the best possible solution?
Or what's the best possible value that we can achieve?
And in a decision problem, we have to ask, is it possible to achieve a value of k?
And so up here the answer will be some number. And down here, the answer can only be yes or no.
And there are two main reasons why we do this.
First of all, it makes our lives much easier in some of the proofs that we're going to dive into.
And secondly, it's also a little bit more accurate because when you talk about
the P versus NP problem, although it's often stated for optimization problems
such as those we have so far discussed, the whole theory
has actually been developed for decision problems.
Now, you're probably thinking what difference does it make to talk
about an optimization problem versus a decision problem?
Or is this going to make any significant difference, for example, with the question of tractability
versus intractability and actually it doesn't really make that much of a difference,
but I'll actually have you figure out the details here in our next quiz.
And I would like you to think about four things and then give me your answer.
So the first question is, if the optimization version of a problem turns out to be tractable,
then what about the decision version of that problem?
Do we know that decision versions for sure to be tractable, intractable, or can we really say?
The second question I have for you is the same thing, only with intractability,
so if we know the optimization version of a problem to be intractable,
then what about the decision version, is that for sure, going to be tractable, intractable,
or is this a case where we can't really make a clear statement having only this information here
that the optimization version is intractable, and finally questions number 3 and 4
are exactly the other way around than questions 1 and 2, so here we have information
about the decision version of a problem, so question number 3,
if the decision version of a problem is tractable, then what about the optimization version?
And question number 4, if the decision version of a problem is intractable,
then what about the optimization version?
So please give your answers here with one of each of the three possible choices here.
