This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
en:dydaktyka:csp:lab2 [2017/03/13 00:41] msl [Problem wielo-plecakowy] |
en:dydaktyka:csp:lab2 [2020/03/13 13:48] (current) msl |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| - | ====== Constraint Programming: Basic Modelling Tenchniques ===== | + | ====== Constraint Programming: Basic Modelling Techniques ===== |
| ===== Symmetry Breaking ===== | ===== Symmetry Breaking ===== | ||
| - | The problem is said to contain symmetry if there are exist classes of equivalent solutions --- solutions, which are called symmetrical because there exist a simple mechanical procedure to obtain one from another. Graph Coloring Problem has a very obvious symmetry --- in every solution we can freely swap colors, e.g. every red node repaint as blue, and every blue node repaint as red. Solutions of this kind aren't bad, just redundant, leading to much bigger search space. Symmetry breaking prunes the search space by removing symmetries from the problem. | + | The problem is said to contain symmetry if there exist classes of equivalent solutions --- solutions, which are called symmetrical because there exists a simple mechanical procedure to obtain one from another. Graph Coloring Problem has a very obvious symmetry --- in every solution we can freely swap colors, e.g. every red node repaint as blue, and every blue node repaint as red. Solutions of this kind aren't bad, just redundant, leading to a much bigger search space. Symmetry breaking prunes the search space by removing symmetries from the problem. |
| + | |||
| + | All files required to solve the assignments are available via [[https://gitlab.com/agh-krr/2019-2020/labs/|the repository]], so clone it first. | ||
| ==== Graph Coloring ==== | ==== Graph Coloring ==== | ||
| * Problem: Same as [[http://ai.ia.agh.edu.pl/wiki/en:dydaktyka:csp:lab1#graph_coloring|before]] | * Problem: Same as [[http://ai.ia.agh.edu.pl/wiki/en:dydaktyka:csp:lab1#graph_coloring|before]] | ||
| * Assignment: | * Assignment: | ||
| - | - Download and extract {{:pl:dydaktyka:csp:csp_coloring.zip|archive}}. | + | - Look at and comprehend ''lab3/graph_coloring/graph_coloring.mzn'' model. |
| - | - Look at and comprehend ''csp_coloring.mzn'' model. | + | - Try to solve the ''basic_data.dzn'' instance. |
| - | - Try to solve the ''csp_coloring_data.dzn'' instance. | + | * You can use the model created during previous classes |
| - | * You can use model created during previous classes | + | * There is a chance, that problem would too difficult to be solved in a reasonable time. |
| - | * There is a chance, that problem would to difficult to be solved in a reasonable time. | + | - File ''data_with_clique.dzn'' includes info about the largest clique in the graph |
| - | - File ''csp_coloring_data2.dzn'' includes info about the largest clique in the graph | + | |
| * ''minColorsNumber'' - size of the largest clique | * ''minColorsNumber'' - size of the largest clique | ||
| * ''maxClique'' - indexes of the vertices forming the largest clique | * ''maxClique'' - indexes of the vertices forming the largest clique | ||
| Line 23: | Line 24: | ||
| * Assignment: | * Assignment: | ||
| - Find a symmetry in the problem. | - Find a symmetry in the problem. | ||
| - | - Download and extract {{:pl:dydaktyka:csp:csp_n_knapsack.zip|the archive}} | + | - Look at and comprehend ''lab3/multi_knapsack/multi_knapsack.mzn''. |
| - | - Look at and comprehend ''csp_n_knapsack.mzn''. | + | |
| * Note :!:: this model is not an example of good constraint model; it's just used to show a common technique to break symmetries | * Note :!:: this model is not an example of good constraint model; it's just used to show a common technique to break symmetries | ||
| * Note use of the ''at_most'' global constraint. | * Note use of the ''at_most'' global constraint. | ||
| - | * There is a defined predicate named ''knapsack'' -> it's the first example of user defined predicate | + | * There is a defined predicate named ''knapsack'' -> it's the first example of user-defined predicate |
| - Run model with the associated data file: | - Run model with the associated data file: | ||
| - | * The problem may be to hard for the solver | + | * The problem may be too hard for the solver |
| - | * | + | - Break symmetry using [[https://www.minizinc.org/doc-2.4.2/en/lib-globals.html#lexicographic-constraints|''lex_less_eq'' global constraint]]. |
| - | - Break symmetry using [[http://www.minizinc.org/doc-lib/doc-globals-lexicographic.html#Ipredicate-lex_less-po-array-bo-int-bc-of-var-set-of-int-cl-x-cm-array-bo-int-bc-of-var-set-of-int-cl-y-pc|''lex_less'' global constraint]]. | + | - Run the new model with the same data file |
| - | - Run new model with the same data file | + | |
| ===== Redundant Constraints ===== | ===== Redundant Constraints ===== | ||
| - | There is a good chance the problem can be defined in more than a one way. Also you may find a set of constraints that is sufficient to define the problem. That's cool, however there can exist so called "redundant constraints"; redundant because they do not have impact on the number or quality of the solutions. The only reason to include them into the model is that they may contain additional info about the structure of the problem, therefore giving solver an opportunity to prune the search space (most of the solver prune the search space by propagating constraints, redundant constraint may boost this process). | + | There is a good chance the problem can be defined in more than one way. Also you may find a set of constraints that is sufficient to define the problem. That's cool, however there can exist so called "redundant constraints"; redundant because they do not have an impact on the number or quality of the solutions. The only reason to include them in the model is that they may contain additional info about the structure of the problem, therefore giving solver an opportunity to prune the search space (most of the solver prune the search space by propagating constraints, a redundant constraint may boost this process). |
| - | ==== Magiczne ciągi ===== | + | ==== Magic Sequence ===== |
| + | * Definition: Same as [[http://ai.ia.agh.edu.pl/wiki/en:dydaktyka:csp:lab1#magic_sequence|before]] | ||
| + | * Assignment: | ||
| + | - Look at and comprehend ''lab3/magic_series/magic_series.mzn'' | ||
| + | - Add redundant constraints, hints: | ||
| + | * what should be equal the sum of the magic sequence? | ||
| + | * what should be equal the sum of sequence elements multiplied by their indexes (if indexing starts from 0)? | ||
| + | - Compare solving time with and without the redundant constraints. | ||
| + | - Smile with satisfaction | ||
| - | Jeżeli ktoś nie zrobił tego wcześniej, proszę przed przystąpieniem do ćwiczenia przynajmniej przeczytać opis problemu na stronie poprzedniego laboratorium. | + | ==== Channeling ==== |
| - | * Należy: | + | If you have more than one model of the same problem, you can combine them into a single model. Why would one do that? Mostly because some constraints are easier to express with different variables. Other reason could be that the second model often makes a great example of the redundant constraints. |
| - | - Ściągnąć i zrozumieć model: {{:pl:dydaktyka:csp:csp_magic_series.zip|}} | + | |
| - | - Dodać do niego nadmiarowe ograniczenia. Podpowiedzi: | + | |
| - | * ile powinna wynosić suma magicznego ciągu? | + | |
| - | * ile powinna wynosić suma elementów magicznego ciągu, pomnożonych przez numery swoich indeksów (indeksując od 0)? | + | |
| - | - Porównać działanie modelu ''z'' i ''bez'' dodatkowych ograniczeń na dłuższych ciągach. | + | |
| - | - Czerpać zyski. | + | |
| - | ==== Modele łączone - n-hetmans return! ==== | + | * Problem: [[http://ai.ia.agh.edu.pl/wiki/en:dydaktyka:csp:lab1#n-queens|N-Queens]] again |
| - | + | * Assignment: | |
| - | Jedną z technik dodawania dodatkowych ograniczeń jest wykorzystanie kilku modeli tego samego problemu równocześnie. Pomimo tego, że wszystkie te modele mogą wskazywać na te same rozwiązania, zawarte w nich ograniczenia w innej kolejności zawężają dziedziny zmiennych. Kombinacja kilku modeli pozwala na ograczenie przestrzeni przeszukiwania z kilku "perspektyw" równocześnie. | + | - Look at and comprehend ''lab3/n_queens/n_queens.mzn'' |
| + | - Add another model of the problem | ||
| + | * try to use the boolean array of variables ''array[1..n, 1..n] of var bool: qb;'' (queen boolean) | ||
| + | * add missing constraints so the second model was also independent | ||
| + | - Channel constraints from the both models: | ||
| + | * create constraint that connects variables from the model | ||
| + | - Compare running time of the normal and channeled model | ||
| + | - Add symmetry breaking to the problem by using ''lex_lesseq'' constraint on the different permutations of the ''qb'' array | ||
| + | * below the assignments there is a code listing with all permutations calculated in MiniZinc, can you tell what symmetries they represent? | ||
| + | - Compare running time again | ||
| + | - Give yourself a [[https://youtu.be/kMUkzWO8viY|self-five]], <wrap lo>in this case, it may not improve the running time, but the technique itself is very useful in more complex problems</wrap> | ||
| - | * Problem: ponownie problem n-hetmanów. | + | <code> |
| - | * Należy: | + | array[int] of var bool: qb0 = array1d(qb); |
| - | - Odkopać model n-hetmanów z poprzednich zajęć. Ewentualnie pobrać ten: {{:pl:dydaktyka:csp:csp_n_hetmans.zip|}} | + | array[int] of var bool: qb1 = [ qb[j,i] | i,j in 1..n ]; |
| - | - Dopisać do niego drugą możliwość wyboru zmiennych decyzyjnych i ograniczeń | + | array[int] of var bool: qb2 = [ qb[i,j] | i in reverse(1..n), j in 1..n ]; |
| - | * Jeżeli ostatnio zmienne były indeksowane kolumnami i wskazywały na numer wiersza, należy dopisać model odwrotny. | + | array[int] of var bool: qb3 = [ qb[j,i] | i in 1..n, j in reverse(1..n) ]; |
| - | - Powiązać ze sobą ograniczeniami oba zestawy zmiennych. | + | array[int] of var bool: qb4 = [ qb[i,j] | i in 1..n, j in reverse(1..n) ]; |
| - | * Pomocne może być globalne ograniczenie [[http://www.minizinc.org/2.0/doc-lib/doc-globals-channeling.html|''inverse'']] | + | array[int] of var bool: qb5 = [ qb[j,i] | i in reverse(1..n), j in 1..n ]; |
| - | - Porównać działanie modelu pojedynczego i dualnego na problemach wielu hetmanów. | + | array[int] of var bool: qb6 = [ qb[i,j] | i,j in reverse(1..n) ]; |
| - | - Czerpać zyski <wrap lo>z nabytej wiedzy, bowiem nowe solvery radzą sobie z n-hetmanami lepiej bez nadmiarowych ograniczeń. Niemniej w innych modelach technika ta może okazać się istotną optymalizacją.</wrap> | + | array[int] of var bool: qb7 = [ qb[j,i] | i,j in reverse(1..n) ]; |
| - | + | ||
| - | ===== Sterowanie przeszukiwaniem ===== | + | |
| - | + | ||
| - | O ile budowanie modelu w programowaniu z ograniczeniami jest zadaniem czysto deklaratywnym, często wiele zależy od sposobu, w jaki przeszukiwana jest przestrzeń rozwiązań. Istnieje wiele możliwych strategii, jednak w podstawowym przypadku można je zdefiniować poprzez dwa parametry: | + | |
| - | + | ||
| - | - kolejność wyboru zmiennych - np. w kolejnym kroku solver ustali wartość zmiennej, która ma najmniejszą dziedzinę; ma największą dziedzinę; w kolejności zdefiniowania przez użytkownika, etc. | + | |
| - | - kolejność wyboru wartości - po wyborze zmiennej, solver musi przypisać jej wartość z dziedziny, np. wartość najmniejszą, medianę dziedziny, wartość losową, etc. | + | |
| - | + | ||
| - | Aby w MiniZincu wymusić na solverze strategię przeszukiwania, należy użyć tzw. adnotacji (ang. //annotations//). W ogólnym przypadku można je przypisywać do zmiennych podobnie jak inne parametry, ale najczęściej używa się ich przy specyfikacji celu, np. <code> | + | |
| - | solve :: int_search(array, first_fail, indomain_min, complete) satisfy; | + | |
| </code> | </code> | ||
| - | daje solverovi znać, że tablica zmiennych array zawiera liczby i należy ją przeszukiwać, najpierw wybierając zmienne o najmniejszej dziedzinie (''first_fail''), następnie przypisywać im jak najmniejsze wartości (''indomain_min'') i przeszukiwać całą przestrzeń rozwiązań (''complete''). | ||
| - | Istnieją bardziej zaawansowane metody określania strategii w poszczególnych solverach, ale MiniZinc chcąc być językiem niezależnym od solvera nie może sobie na to pozwolić. Aktualnie konstrukcje pozwalające na pełną kontrolę nad procesem przeszukiwania działają jedynie dla solvera Gecode i wymagają dodania pewnych patchy do standardowej dystrybucji MiniZinc. | + | ===== Reified Constraints ===== |
| - | ==== n-hetmans again ==== | + | [[https://en.wikipedia.org/wiki/Reification|Reification]] in Constraint Programming means treating the constraint as a first-order citizen, i.e. you can use the constraint as a boolean value in your model. If you've used the ''bool2int'' function in the Magic Sequence problem, you could do that only because the constraint ''='' has been reified. Reification allows us to create models with "soft constraints" or "conditional constraints", i.e. one constraint has to be satisfed only if the second one is satisfied too, otherwise they both can be ignored. To do that, one has only to reify the constraints and connect them with the implication: ''constraint1 -> constraint2''. Let's practice this quite useful technique :) |
| - | + | ||
| - | * Problem: już chyba każdy go zna. | + | ==== Stable Marriage Problem ===== |
| - | * Należy: | + | * Problem: There are two classes of objects (men and women, for example) that have to be matched according to their preferences. We say that a matching (marriage) is unstable if both spouses would prefer to be with somebody else. You can read more about this problem on [[https://en.wikipedia.org/wiki/Stable_marriage_problem|wikipedia]]. |
| - | - Spróbować różnych adnotacji do przeszukiwania dla tego problemu, np. <code> | + | * Assignment: |
| - | int_search(rows, input_order, indomain_min, complete); | + | - Look at and comprehend ''lab3/stable-marriage/stable-marriage.mzn'' |
| - | int_search(rows, input_order, indomain_median, complete); | + | - Add missing variables, constraints |
| - | int_search(rows, first_fail, indomain_min, complete); | + | - Give a high-five to your teacher :) |
| - | int_search(rows, first_fail, indomain_median, complete);</code> | + | |
| - | - Zerknąć do [[http://www.minizinc.org/downloads/doc-latest/flatzinc-spec.pdf|dokumentacji flatzinc, rozdział 5.6.1]]. Przeczytać opisy różnych strategii i wypróbować jakąś szaloną. | + | |
| - | - Porównać prędkość działania solvera n-hetmanów dla różnych strategii. | + | |
| - | - Wyczerpać zyski z dzisiejszego laboratorium. | + | |
