Basic Modeling for Discrete Optimization - Third Model by The University of Melbourne #3

Key Takeaways

nube Guan Yu and Zhang Fei wanted to borrow horses from a wealthy man Jung shrew King to help suppress the Yellow Turban Army juncture ping wondered if the brothers were talented enough to make good use of his horses for the suppression he decided to give them a problem and promised to let them borrow the horses if they could solve it the problem was to color a map of the Han Dynasty using at most four colors such that the adjacent states didn't share the same color nube thought for quite a while about the problem but was unable to solve it so urgent was the need to borrow the horses that he sought help using the magical tablet so our three heroes are interested in building a stronger army and so they go to visit junk shipping in order to see if he'll sponsor them for horses but Jang souping is a wise man and he sits them a challenge first to show that if he's going to sponsor them then they're going to show the capabilities is going to be able to defeat the Yellow Turban rebels so he gives them this map of the Han Dynasty and says for color of this map so let's look at a music model to call this map so here we have an enumerated type color which is made up of four different colors green blue pink and yellow and each of the different provinces is declared as a variable of this numerator type color so the province suit we're going to have to decide which color to color that province and the same all the rest of the provinces now we have to write down the constraints of the map so any two provinces which are next to each other we may have to make sure that they have a different color and so we've written down all those constraints you can see they're so many that we've gone to a very small font size let's just look at a few of them so the province yang is next to the problems young and so we've written this constraint here the constraint yang is not equally um and similarly the province young is next to problem G so I've written that constraint young is not okay ye the province young is next to the problem Jing so I've written that constraint young is not agreeing and the problems young is next to su and so we've written this constraint youngest article su and we write down all the rest of these neighbour relationships between the different provinces with our sole satisfy we're just trying to find one particular solution to the map coloring problem and we can run this music moral and will result in their coloring able to print out all the different colors of the map knows we didn't need an output function and we're just going to get for each particular province which color it got and if we look at the result we'll see a correct coloring of the map where every province is colored different to every neighbor so what we've seen here is a new feature of meaning which is a numerator types so enumerated types are a way of defining a finite set of named objects and then we can make decisions and parameters can be then about these enumerated types and array indices which will introduce later can also be over there's a numerator types and we can also choose sets over these enumerated types which we'll see shortly so they're declared by this enum and in the name of the enumerator type and then they're defined by this equality and numeron name is then then given this set of identifiers so we gave the example of the set of different colors that we were given and then we can declare the variables by this VAR and then the name of the numerator type and then the variable name or if we don't have the VAR here then it's a parameter and we're just picking a particular fixed version of this enumerated type so we could modify our program to have a different enira type we just say let's color the graph with green blue and pink so can we color this map with just three colors and we could run our music model then we will result in this unsatisfiable right so this is a new about but we haven't seen before and it should be fairly clear what it means it means that there is no way of free coloring the map of the Han Dynasty provinces so we know if you know that there is a four color map theorem says there's always a way of for coloring a planar map like this one but there's no way of coloring this map with three colors and so our music our solver is telling us there is no possible way of solving this problem and getting back the result unsurvivable so in this lecture we've introduced enumerated types which is a way of introducing a named set of objects so their real importance is that they introduce a notion of type safety of models if I have two different sets of objects that I'm making decisions about then I don't want to mix them up I don't want to make decisions about this one when I meant to be making decisions about that one so new motor types will in fact map down in the solvers to be integers we could have used the numbers one to four to a color out map instead of these color names and in fact the numbers one to four is what the solver will see when it sees this problem but by using this name type we don't only get a model which is easier to understand but we also get this notion of type safety which will help us later on with more complicated models we also saw for the first time an unsatisfiable model we have to be clear that not every model has a solution so sometimes we'll ask to find a solution that will be none more often and unsatisfiable model will arise because we typed in a model incorrectly there's a bug in our model or bug in our data and unsatisfiable will be a trigger for us to try and find out what's going wrong and the problem we solved here was graph coloring this is a very classic problem in graph theory and it applications in register allocation and compilers and timetabling lot of other things and of course usually as I'll usually say pure graph coloring is better handled by specialized algorithm so lots of problems in graph theory have been studied intentionally like graph coloring and if you're doing a pure graph coloring problem then you should use a specialized algorithm but if we add some side constraints about how specific parts of the graph should be colored then using a general discrete optimization like mini Zink is the way to go you