Okay guys. You know, I talked about the language of

constraint programming the last time and that I was presenting an overview of what

it is. You know, I lied.

Okay. So there is one more thing that I need

about, that I need to talk to you about and this is global constraints.

Okay? So global constraints are one really

important feature of the language of constraint programming.

They are part of this richness and the ability to express complex substructure,

okay. They are a critical path of any

constraint programming system and this is a very active, you know, research area in

the field, okay. So, what is the global constraints?

The global constraints capture the, the, the combinatorial substructure arising in

many applications. So, the good analogy here.

Is essentially Lego blocks. Okay?

So you, when, when you buy Legos, you have a big set of pieces that are

encapsulate things that you want to build from.

Global constraints are exactly like that. They are capturing substructure that are,

that are arising in a variety of application.

And we can use them directly. We don't have to express them in terms of

smaller blocks. Okay?

They make modeling, you know, easier, they are more natural.

They make the model easier to read, they make the model more concise.

But, the real, real interesting aspect of them is also the fact that these

combinatorial substructure are really given to the solver.

We capture this structure and we give them, you know, directly to the solver.

And then the solver has a lot of information to do interesting things.

And in particular, it gives the solver the ability to use dedicated algorithm

for each one of these substructure. So I'm going to give you examples.

First, you know, I'm going to show you the modeling and then I'm going to tell

you how these things can actually help you from a computational standpoint.

So, what you see here is probably the most famous, you know, global

constraints, the alldifferent constraints.

And these constraints is basically saying that the variable x1 to xn have to be

given a value, have to be given all different values.

So no two variables in x1 and xn can, can take the same value.

Okay? Now remember, you know, we had this

beautiful queens example in the previous lectures?

Okay? And what that was basically saying is

that all these queens, okay, have to be placed on the chessboard.

Such that they were not on the same row, the same upward diagonal and the same

downward diagonal. Okay?

Well, you can essentially express the same example with three constraints and

that's what you see there, okay. Same decision variable but now instead of

saying all these rows, you know, in which the, you know, the queens were placed,

had to be, you know, pairwise different. You just specify one constraint which

says all the rows have to be different. Okay.

That's what you said. That's what you are stating there.

Now, if you look at the other constraints, these guys, okay.

So what you see once again is that, you know, you see row i plus i, row j plus j.

Essentially what it is that you have a bunch of expression row i plus i row j

plus j row k plus k and essentially all these things have to be different and

that's what you express in this constraint.

Okay? And so that's once again a-alldifferent

constraints. Okay?

And these all different constraints is going to capture the upper diagonal and

you'll have the last alldifferent constraint which is going to capture the

lower bound. Now you see that this operator all over

there, essentially this is [INAUDIBLE] comprehension, okay.

So essentially what it says is take all i in r, okay?

Take the, the collection of expressions that you see over there, which is in this

particular case, you know, row i plus i, and collect all these in an array, and

then that's, that's the array that you have as a result, okay?

In this particular case, this array's going to be passed to the alldifferent

constraints, okay? So, essentially, this is collecting a

bunch of things, and then saying that all these variables, or expression, in this

particular case, have to be different. Okay, so in a sense we can express this

screen problem with three constraints. Okay?

Now why would we ever do that? It seems more complicated and so on but

what I'm going to show you is that this has a lot of computational advantages.

Okay, so let's look. You know, you know from now on from, you

know, previous lecture that every constraints has two goals in life.

The first goal is to, is to do feasibility testing.

You want to know if the constraints is feasible.

Okay? Remember we had a constraints.

We have essentially for every one of the variables the domain.

Okay, so that's a set of values that a variable can take.

Okay? So, we often use this notation here, you

know, this is a constraint and then we said that d1 is the domain of x, of x1

and then d2 is the domain of x2 and so on.

Okay? So, what is feasibility?

Feasibility is finding a value in this domain such that when the variables are

assigned it's variables the constraint is satisfied and that's essentially what

this formula is basically telling you. Does there exist a value in the domain of

x1? Okay?

Such that, and there, there, such that if I, so that I can find another value for

the you know, v2 in the domain of x2, and v3 in the domain of x3.

Such that, when I assign all these values to the cons, to the variable of the

constraint, the constraint is true. That's feasibility testing.

Okay? Now, let's look at this constraint here,

okay? So, we have the alldifferent, okay, which

is on three variables, x1, x2, x3. You know, cannot be much simpler than

that, okay? And then, for every one of these

variable, okay, so, we have a domain. And in this particular case, three

variables with a very simple domain, value 1 and value 2.

Okay? Now, so in a sense, all different and

then these other domains. Okay?

Is this feasible? Okay.

No, it's not. Okay.

Why is not feasible.Okay. It's not feasible because we have three

variables and we have two values. Okay?

So people in computer science code, that the pigeon hole principle, I have never

seen a pigeon in a hole but anyway. So this is the pigeon hole principle.

Probably pigeons are looking for holes and if you have three pigeons and two

holes, they can't all fit in the hole. Okay, in the holes.

Okay, so is, in a sense here what it's telling you, three variables two value,

impossible. You can't have feasibility.

Right? So we know that the global constraints is

going to say no, no, no, this is not feasible.

Okay. Now, if we look at, you know, the first

formulation that we had when we were using the queen's example we had

essentially three constraints to express the same thing.

Right? So we said x1 can not be equal to x2.

X2 cannot be equal to x3 and that x3 cannot be equal to x1.

Now if you look at every one of these constraints independently.

Okay. Do the same test that we just did for the

that alldifferent. What do you get?

Well. For the first constraint I can take the

value 1 for x1 and the value 2 for X2 and this constraint is fine.

It's satisfied. Okay.

You look at the second constraint and you can do exactly the same trick.

Of course this time is the value x2 and the value x3.

Okay? But essentially it's true as well.

And the third constraint is true as well. So all these constraints, when you look

at them independently, they are fine. Okay?

So, the system is going to say, yeah, you know, the set of constraints look

satisfiable. Okay?

Although we know for sure that they are not.

Okay? So, that's one of the big impact of

global constraints. Okay?

They can actually detect infeasibility earlier in the search space.

Therefore, they make your search space smaller, and your search more efficient.

Okay? So, in a sense if you think about it,

this is the computation model of constraint programming.

Right? So, you have this domain store, it has

the domain of the variables. And gravitating around if you have all

the constraints. These constraints only talk to the domain

store. They don't talk about each other.

And because of that, if you handle them independently, they don't necessarily

communicate. Right?

So, they can talk to the same domain, and that's what you see.

Right? So this is the first constraint, talking

to this two domain. The second constraints have a domain in

common, and another one. The two constraints have again another

domain in common and another one. And so you see that they are all linked.

The path can be long, but they are all linked.

But when you reason about them independently, you lose that length.

Okay? What the global constraints is saying,

okay, so let's encapsulate all these constraints inside one global

constraints, which talks to all the these domains at the same time and now I can

actually detect infeasibility. Okay?

So, so now we have done only the first thing that a constraint does for a

living. Right?

Which is testing feasibility. The other thing a constraint does for

living is pruning. We want to look at these domains and see

if we can knock down some of these values, okay?

Because that reduces search space, also make the search more effiecient.

Okay, so the question that we have is given a variable a value vi inside the

domain of variable x i. Okay, can I use sin, the value of vi to

xi and still find the solution to that constraint.

Okay? So in a sense what that means is that I'm

assigning vi to xi but I want to find out if that value seems dominant of the other

variables such that the overall constraints is going to be

satisfied.Okay? So how do I do that?

This is the formula right? So I'm assigning xi to the i, okay?

And then I want to find a value v1 for v1.

Value v, i minus 1. Inside, you know?

For, for the variable xi minus one. And so on and so forth, such that this

constraint is true, okay? So I pick up one variable, one value.

And then I see if I can verify find values in the other domain of the

variable. In the domain of the other variable such

that the constraint is satisfied, okay? So look at this, okay?

So this is once again alldifferent constraints.

Now I have three domains, one two for variable x1, one two for variable x2, and

then one through three for variable x3, okay?

And I'm asking you, can I actually prune the search space, okay?

And the answer is, yes I can, okay? Why?

Well, we know because of this beautiful pigeonhole principle, that if I look only

at x1 and x2, I have two variables, and they can only take two value, 1 and 2,

okay? Therefore, if x3, the friendly x3,

actually take either 1 or 2. No, we won't find a solution, okay.

So what you have to, what you can deduce easily, is that x3 cannot be 1 and cannot

be 2 it has therefore to be 3, okay. And therefore, you reuse the search

space, you know. X1 and x2 can take either 1 or 2,

provided they have a different values, okay.

So the only pruning that you can get is remove the, removing the value from 1 and

2. Okay?

Now, that's the pruning that we can get. But once again, if you don't express the

global constraint, if you use the composition here in terms of pair wise

constraints, you would not prune anything.

Okay? Because essentially, for x3, you know,

when you look at x3. X3 and x1, well they can, you know, x, x3

can take the value 1 and x2 and x1 can take the value 2 and that's fine.

X3 can take the value 1 and x2 can take the value 2 and that's fine as well.

So you never detect that you actually have to remove the value one and two from

x3. So the second big benefits of global

constraints. Not only do they detect feasibility

sooner, but they also actually, make it possible to prune the search space more.

They remove value from the domain of the variable.

And why is that important? Because now the domain are smaller

another constraint can come in and do more reasoning and maybe find an

inconsistency or maybe prune new values from the domain which will in, in turn

probably find some inconsistency or some more values from the domain, okay.

So, this fixed point is very important. That's why you remove the search space as

much you can. Okay?

So the million dollar question though is what?

Can I actually, you know, detect feasibility and achieve this optimal

pruning for this global constraint? Okay.

And the answer to that sometimes we can, sometimes we cannot.

Okay. So sometimes we will able, we will be

able to detect feasibility and I will show you examples of that.

And sometimes you will have to relax your standards.

Okay, which basically means that you won't be able to prune everything or you

won't be able to detect insatisfiability all the time.

but, but you will still get some pruning and detect visibility early, okay.

the other thing that you can say is I'm going to sacrifice computation time and

essentially some of these algorithms and one of my students became an expert in

finding exponential algorithm from pruning constraints, okay?

And what, what you do there is you sacrifice time but you will have the

optimal pruning. You will detect infeasibility as soon as

you can, okay? So the answer here is sometimes some

global constraints you will be able to deal with very efficiently, sometimes you

will have to relax your standards. Okay, now I told you when, you know, in

the advertisement of this class that, you know, you will never have to solve a

sudoku in your life, okay? And I'm going to keep that promise, okay?

And so this is a sudoku and we want to solve it using constraint programming.

Okay? So this is a very simple model on how to

do that. Okay?

So the decision variables are very easy right?

So you look at everyone of these square, little square everywhere.

You know, there will be a decision variable associated with that and that

decision variable will be the number which is associated to that square.

Okay? The digit.

Okay? So essentially that's what you see there.

You know these are the decision variables over there.

Okay, they are basically [UNKNOWN] a matrix of, of, of variables, and these

variables take the value between one and nine.

Okay? Now, the constraints of these problems,

you will have different types of constraints.

The first set of constraints will be the fixed position.

Okay. So when you look at this thing, there are

a bunch of fixed positions all over the place.

We'll have to fix these values, these variables to these values.

And then once again, we have only three types of constraints, okay.

Well, actually one type of constraint but three different ways of stating that,

okay. So we have only alldifferent constraints,

but some of them are going to be on the rows, some of them are going to be down

the column and some of them are going to be on the squares, okay.

So when you look at the first one. Essentially that constraint, is basically

stating, that all the values on a particular row have to be different.

The second one is basically saying that all of the values in a column have to be

different. And the third one is basically, you know,

you look at the square, you know, three by three there, and you are expressing

that all these values have to be different.

These are the constraints of the Sudoku and that's all you have to do.

So this simple program. Is why is at least part of the reason why

I would never do a sudoku in my life, right?

You know it's so simple, right? So the next thing that I want to show you

is actually this is a very efficient way of actually solving this thing.

Okay, so look at this puzzle. Okay so the first time I run the program,

I traced it on this particular example. It deduced that, of course it, it fixed

all these position, and then it deduced that this value has to be 2, okay?

Now, that's very interesting, okay? So, I was like how did it do that?

Okay? And so you have to look at a couple of

things, okay? So, you see, you know, there is there is

a 2 over there. Right?

So, which basically means that I cannot put a 2 over here, okay?

There is no way I can do that. Okay, and then you have a two over here,

which basically means it can't have any two on this last line, okay?

So the tree position where I can actually put a two is here and these two gaps,

right? Okay?

So but look at, look at the, the square here in the middle.

Once again, I know that I have a two and therefore, I cannot put a value over

there, okay? Now I also know, what do I know?

Okay. So I know that I have a 2 over here.

So these guys here, cannot get a value of 2.

So the only two positions where I can get a 2 here, is these two guys.

Okay. These 2 guys can take a two.

I don't know which one, right? Okay?

But at least one of them has to take a 2. Therefore what I know is this guy, cannot

have a 2 over here. Okay?

And therefore, the last position at which it's get a, get a 2 is over there.

That's what the system did. That's what this pruning for the all

different constraints were able to do. It's magical, right?

Okay, so if you do that, then it's going to know, going to continue deriving

all these values all over the place. That's what it gets, just propagating

these constraints, okay? And then, the system will make a choice.

It will assign the value 5 at the top over there.

Choop! Here.

And that it will make more deduction than when you assign one more value over here,

okay? So there's a value of 4.

And then propagate and find a solution, and that's it.

Two choice, no back-track, solution immediately.

Okay. Now you have to know that sudokus are

easy for us, and the reason is that they have to be easy for human, which

basically means that human don't like to make choices, so these sudoku are

actually designed so that you can mostly deduce the solution and don't go into

exploring a huge tree. That's why these things are really easy

for constraint programming in general. Okay, so, so we have seen now essentially

global constraints and global constraints are these very important area of

constraint programming. There are a lot of people actually

exploring this. Okay?

And, I want, I want to show you one more type of constraint which is the table

constraint. And it's probably the simplest global

constraint, okay? And so this is an example for you to

understand. I have three variables okay?

x, y, and z, okay? This is the domain of the variable 1, 2

1, 2, and then 3 ,4 ,5 for, for z. Okay?

And then essentially one of the things we can think of is, you know, what are the

possible combination of all these variables, okay?

What are the total possibilities? And essentially there are 12 of them,

okay? So you could see each product between,

you know, the domain of x, the domain of y, and the domain of z, okay?

So that's what we have, okay? So in this particular case there are 12

possibilities. So what a table constraints is doing, is

actually specify a subset of these possibilities as the legal, you know,

assignment of these variables. So, in this particular case, we have four

of them, okay. So you see that the first one is 1, 1, 5.

X is equal to 1, Y is equal to 1, and Z is equal to 5.

The second one is 1, 2, 4. That's what you see here, okay.

And so in this particular case, the legal combination is 1 for x, 2 for y, and 4

for z. Okay.

And so once again what is interesting to see is what happens when you get more

information. Okay.

So assume that I have more information about z.

What's going to happen? Well, I know that z cannot be equal to 5.

What does that mean? Okay, so the value z there is not legal

anymore. I have to remove that combination.

I have to remove all the combinations where I have actually z is equal to 5.

There is only one here, okay? And now, essentially, I look at the, I

look at the other variables, and what do I see?

Look at this guy, look at y. The only value which is left in the

column of y is 2. So I can immediately deduce that, in a

sense, y has to be 2 and the value, you know, and this is what this reflect here,

okay? So, so, in a sense, y has to be different

from y, it can only be 2, okay? So that's essentially the table

constraints. Once again, it's a very useful

constraint. It always specify a subset of a cartesian

product. For all the variables.

Okay? So let me conclude this lecture by one

more thing, okay? which is how can we find optimal solution

in using a constraint programming? So remember we discuss graph coloring or

map coloring actually In the first two lectures, okay?

And so what, what I'm doing here is generalizing a little bit of that

example, okay? And instead of using colors, I'm using a

number from 1 to 4, and what I'm going to do now is not only color this country

such that they get a different color, in particular, in this particular case a

number between 1 and 4. But I also want to minimize the number of

colors that I'm using. Okay?

So basically here, I'm basically saying, okay, minimize, okay, what the maximum

color that this variables can subject to the fact that two adjacent countries have

to be colored differently, okay? So, I can do that.

Okay, so that's a model which is essentially optimizing the number of

colors that I have to use. In all the possit-.

Selecting essentially the solution, the feasible solution, which uses the, the

fewest colors, okay? How do I do that?

In constraint programming, as I told you, the focus is on feasibility.

And what you do when you're trying to optimize is you're trying to reduce

optimization problem to feasibility. You essentially solve a sequence of

feasibility problems. You find a solution and then you put an

additional constraint which says, oh, but the next solution is to be better than

the one that I just found. Okay so when we find a solution with four

color we are going to say I want to find a solution which you're only using three

colors. Okay, so we are guaranteed to find an

optimal solution. You know, theoretically.

You know, in practice it may take too long, okay and this is going to be a

strong method when essentially the constraints are too hard, that you add as

essentially a strong pruning, okay, and this happens in a variety of problems in

results, allocation and scheduling, and we'll come to these problems at some

point in this class. Okay.

Thank you. That's it for today.

CP 4 - global constraint intuition, table constraint, sudoku

Discrete Optimization

Meet the Instructors