Tired of solving Sudokus by hand? This class teaches you how to solve complex search problems with discrete optimization concepts and algorithms, including constraint programming, local search, and mixed-integer programming. Optimization technology is ubiquitous in our society. It schedules planes and their crews, coordinates the production of steel, and organizes the transportation of iron ore from the mines to the ports. Optimization clears the day-ahead and real-time markets to deliver electricity to millions of people. It organizes kidney exchanges and cancer treatments and helps scientists understand the fundamental fabric of life, control complex chemical reactions, and design drugs that may benefit billions of individuals. This class is an introduction to discrete optimization and exposes students to some of the most fundamental concepts and algorithms in the field. It covers constraint programming, local search, and mixed-integer programming from their foundations to their applications for complex practical problems in areas such as scheduling, vehicle routing, supply-chain optimization, and resource allocation.
CP 4 - global constraint intuition, table constraint, sudoku
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Do curso por The University of Melbourne
Discrete Optimization
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Constraint Programming
Constraint programming is an optimization technique that emerged from the field of artificial intelligence. It is characterized by two key ideas: To express the optimization problem at a high level to reveal its structure and to use constraints to reduce the search space by removing, from the variable domains, values that cannot appear in solutions. These lectures cover constraint programming in detail, describing the language of constraint programming, its underlying computational paradigm and how it can be applied in practice.
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Professor Pascal Van Hentenryck
Dr. Carleton CoffrinAdjunct Lecturer
Computing and Information Systems
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.
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