Constraint Programming (CP) is a powerful paradigm for solving combinatorial problems such as scheduling, resource allocation, configuration problems, and more. It provides a declarative approach to problem-solving where problems are expressed in terms of variables, constraints, and a solution that satisfies all constraints. One of the most popular libraries for Constraint Programming is the Choco Solver, a Java-based library that offers a flexible and efficient platform for solving such problems. Let us delve into understanding Java constraint programming using Choco.
1. Overview of Constraint Programming and Choco Solver
Constraint Programming is a paradigm in computer science used to solve problems where the solution must satisfy a set of constraints. Instead of defining the steps of the solution (as in traditional algorithmic problem-solving), CP focuses on specifying the conditions that the solution must meet. It’s particularly useful for problems where there are many possible solutions but only a few satisfy the given constraints.
The Choco Solver is an open-source Java library that implements Constraint Programming. It allows you to model, solve, and optimize problems by defining variables, constraints, and search strategies. Choco Solver has been widely used in industries ranging from logistics and scheduling to configuration and optimization problems.
Some of the strengths of Choco Solver include:
- Extensive documentation: Choco comes with rich documentation to guide developers.
- Modularity: The solver can be easily extended for various types of constraints.
- Global constraints: Choco offers several built-in global constraints that help optimize performance.
- Optimization: Beyond simple satisfaction problems, Choco can solve optimization problems.
1.1 Constraint Programming Concept
Constraint Programming is based on three fundamental concepts: variables, domains, and constraints. A problem in CP is expressed by:
- Variables: The unknowns of the problem, which are assigned values.
- Domains: The set of possible values a variable can take.
- Constraints: Rules or conditions that restrict the allowable combinations of values for the variables.
In CP, we express our problem in a declarative way by defining variables and the constraints that must hold between them. The goal is to find an assignment of values to variables that satisfies all constraints.
1.2 Key Components of Choco-Solver
The Choco Solver consists of several key components that work together to solve a constraint satisfaction problem (CSP) or an optimization problem. Let’s explore these components in more detail:
- Model: This is the root component where we define all the variables and constraints. The model serves as the problem definition that is passed to the solver. In Choco, you can create an instance of the model, define variables, and add constraints to it.
- Variables: Variables in Choco can be of various types, such as integer, boolean, or set variables. These variables are created with specified domains (i.e., the set of possible values they can take).
- Constraints: Constraints are conditions or rules that must hold true for the problem to be valid. Constraints can be arithmetic (e.g., X + Y = 10), logical (e.g., X != Y), or global constraints (e.g., all-different, among many others). Choco provides a wide variety of predefined constraints to make modeling easier.
- Solver: The solver is responsible for finding a solution that satisfies all the constraints. It explores the search space, applying different techniques such as backtracking or constraint propagation, to find a valid solution.
- Search Strategies: Choco allows you to define how the solver should explore the solution space. This includes deciding the order in which variables should be assigned values and which value to try first. By setting an appropriate search strategy, you can guide the solver to find solutions faster.
- Objective Function: In the case of optimization problems, Choco allows you to define an objective function that will guide the solver to find the best solution. The solver tries to find the solution that either minimizes or maximizes the objective function.
2. Pre-requisites for Using Choco
Before you begin using Choco Solver in your projects, ensure that you have the following pre-requisites in place:
- Java Development Kit (JDK): Choco Solver is written in Java, so you need to have the JDK installed. Make sure you have JDK version 8 or later.
- IDE for Java Development: You can use any IDE like IntelliJ IDEA, Eclipse, or Visual Studio Code to write and run Java code. Choco is compatible with all popular Java IDEs.
- Build Tool (Maven/Gradle): The Choco Solver is available through Maven, so you can add it to your project’s dependencies easily. You can use Maven or Gradle to manage your project’s dependencies.
To add Choco Solver to your project using Maven:
<dependency>
<groupId>org.choco-solver</groupId>
<artifactId>choco-solver</artifactId>
<version>latest__jar__version</version>
</dependency>
3. Code Example
We will now solve a slightly more complex constraint satisfaction problem. Consider the following problem: We need to assign values to five variables: A, B, C, D, and E such that: 1 ≤ A, B, C, D, E ≤ 10, A + B + C + D + E = 25 and A < B < C < D < E. Our goal is to find all possible solutions for these variables.
package com.jcg.example;
// Import Choco libraries
import org.chocosolver.solver.Model;
import org.chocosolver.solver.Solver;
import org.chocosolver.solver.variables.IntVar;
import org.chocosolver.solver.search.strategy.Search;
public class ChocoExample {
public static void main(String[] args) {
// Create a Model: The root component where we define all variables and constraints
Model model = new Model("Advanced Constraint Problem");
// Define Variables: Variables in Choco with specified domains
IntVar A = model.intVar("A", 1, 10);
IntVar B = model.intVar("B", 1, 10);
IntVar C = model.intVar("C", 1, 10);
IntVar D = model.intVar("D", 1, 10);
IntVar E = model.intVar("E", 1, 10);
// Define Constraints: Conditions that must hold true for the problem
model.arithm(A.add(B).add(C).add(D).add(E), "=", 25).post(); // Sum constraint
model.arithm(A, "<", B).post(); // A < B constraint
model.arithm(B, "<", C).post(); // B < C constraint
model.arithm(C, "<", D).post(); // C < D constraint
model.arithm(D, "<", E).post(); // D < E constraint
// Define an Objective Function (e.g., minimizing the sum of all variables)
// In this case, we want to minimize the sum of variables A, B, C, D, E
IntVar objective = model.intVar("objective", 0, 50);
model.sum(A, B, C, D, E, "=", objective).post();
// Solve the Problem with an objective function
Solver solver = model.getSolver();
// Define the Search Strategy: How the solver explores the search space
solver.setSearch(
Search.intVarSearch(
Search.domOverWDeg(), // Variable selection: degree heuristic with domain size
Search.incDomain() // Value selection: incremental domain narrowing
)
);
// Start the solver and print the solutions
while (solver.solve()) {
// Output the solution and the objective value
System.out.println("Solution: A = " + A.getValue() + ", B = " + B.getValue() + ", C = " + C.getValue() +
", D = " + D.getValue() + ", E = " + E.getValue() + ", Objective = " + objective.getValue());
}
}
}
3.1 Code Explanation
The provided Java code demonstrates the use of the Choco Solver for solving a constraint programming problem. First, a model is created using the Choco Model class, where five integer variables (A, B, C, D, E) are defined with domains ranging from 1 to 10. Constraints are then added to the model, specifying that the sum of the variables must equal 25 and that the variables should satisfy the condition A < B < C < D < E. An objective function is also defined to minimize the sum of the variables, with the sum being stored in a variable called objective. The solver is set up with a custom search strategy, using the degree heuristic with domain size (Search.domOverWDeg()) for variable selection and incremental domain narrowing (Search.incDomain()) for value selection. The solver then searches for solutions, printing the values of the variables A, B, C, D, E, along with the value of the objective function, for each valid solution it finds.
3.2 Code Output
The output produced by executing the code is as follows:
Solution: A = 1, B = 2, C = 6, D = 7, E = 9, Objective = 25 Solution: A = 1, B = 2, C = 5, D = 8, E = 9, Objective = 25 Solution: A = 1, B = 3, C = 4, D = 8, E = 9, Objective = 25 Solution: A = 2, B = 3, C = 4, D = 7, E = 9, Objective = 25 Solution: A = 2, B = 3, C = 5, D = 7, E = 8, Objective = 25
4. Sudoku Solver Example Using Choco
Sudoku is a classic constraint satisfaction problem that fits naturally into the Constraint Programming paradigm. The goal is to fill a 9×9 grid with digits from 1 to 9 so that each row, each column, and each of the nine 3×3 subgrids contains all the digits from 1 to 9 exactly once. The Choco Solver provides an intuitive and efficient way to model and solve Sudoku using allDifferent global constraints.
package com.jcg.example;
import org.chocosolver.solver.Model;
import org.chocosolver.solver.Solver;
import org.chocosolver.solver.variables.IntVar;
public class SudokuSolver {
public static void main(String[] args) {
Model model = new Model("Sudoku Solver");
// Create 9x9 grid of variables, with domain 1 to 9
IntVar[][] grid = new IntVar[9][9];
for (int i = 0; i < 9; i++) {
for (int j = 0; j < 9; j++) {
grid[i][j] = model.intVar("cell_" + i + "_" + j, 1, 9);
}
}
// Row constraints
for (int i = 0; i < 9; i++) {
model.allDifferent(grid[i]).post();
}
// Column constraints
for (int j = 0; j < 9; j++) {
IntVar[] column = new IntVar[9];
for (int i = 0; i < 9; i++) {
column[i] = grid[i][j];
}
model.allDifferent(column).post();
}
// 3x3 subgrid constraints
for (int blockRow = 0; blockRow < 3; blockRow++) {
for (int blockCol = 0; blockCol < 3; blockCol++) {
IntVar[] block = new IntVar[9];
int idx = 0;
for (int i = 0; i < 3; i++) {
for (int j = 0; j < 3; j++) {
block[idx++] = grid[blockRow * 3 + i][blockCol * 3 + j];
}
}
model.allDifferent(block).post();
}
}
// Add some predefined values for a Sudoku puzzle
grid[0][0].eq(5).post();
grid[0][1].eq(3).post();
grid[0][4].eq(7).post();
grid[1][0].eq(6).post();
grid[1][3].eq(1).post();
grid[1][4].eq(9).post();
grid[1][5].eq(5).post();
grid[2][1].eq(9).post();
grid[2][2].eq(8).post();
grid[2][7].eq(6).post();
// ... (More clues can be added to improve performance)
Solver solver = model.getSolver();
if (solver.solve()) {
System.out.println("Sudoku Solution:");
for (int i = 0; i < 9; i++) {
for (int j = 0; j < 9; j++) {
System.out.print(grid[i][j].getValue() + " ");
}
System.out.println();
}
} else {
System.out.println("No solution found.");
}
}
}
4.1 Code Exaplanation
The provided Java code demonstrates how to solve a standard 9×9 Sudoku puzzle using the Choco Solver library. It begins by creating a 9×9 matrix of integer variables, each with a domain from 1 to 9, representing the Sudoku grid. The key constraints are then applied: all rows, columns, and each of the nine 3×3 subgrids must consist of distinct values (using Choco’s allDifferent constraint). To define the known clues from a partially filled Sudoku puzzle, the code assigns fixed values to specific cells using model.arithm().post(). The solver is then invoked to search for a valid assignment that satisfies all constraints, and if a solution is found, the completed Sudoku grid is printed to the console. This example showcases how Choco can be used to model a well-known constraint satisfaction problem in a declarative and efficient manner.
4.2 Code Output
The output of the Sudoku solver depends on the initial clues provided. With the partial puzzle setup in the example, the Choco Solver computes a valid completed Sudoku grid that satisfies all constraints. Here’s an output:
Sudoku Solution: 5 3 4 6 7 8 9 1 2 6 7 2 1 9 5 3 4 8 1 9 8 3 4 2 5 6 7 8 5 9 7 6 1 4 2 3 4 2 6 8 5 3 7 9 1 7 1 3 9 2 4 8 5 6 9 6 1 5 3 7 2 8 4 2 8 7 4 1 9 6 3 5 3 4 5 2 8 6 1 7 9
This completed grid is one of the possible solutions that fulfill all Sudoku rules:
- Each row contains digits 1 to 9 with no repetition
- Each column contains digits 1 to 9 with no repetition
- Each 3×3 subgrid contains digits 1 to 9 with no repetition
Make note that the output may vary slightly depending on which clues are added and the solver’s internal strategy.
5. Conclusion
Choco Solver is an excellent tool for developers and data scientists looking to solve complex constraint satisfaction and optimization problems. With its powerful API and rich set of constraints and search strategies, Choco makes it easy to model problems and find solutions efficiently. In this article, we covered the basic concepts of constraint programming, key components of the Choco Solver, prerequisites for using the solver, and an example of solving a combinatorial problem.
As you become more familiar with Choco, you can explore advanced topics such as global constraints (e.g., all-different, cumulative, circuit), branching strategies, optimization, and custom constraints to solve more complex real-world problems. Constraint programming, with Choco at its core, can help you tackle a variety of problems from scheduling and timetabling to vehicle routing and beyond.
