### Summary

At a glance, Constraint Programming is a method in computer science used to solve complex problems. It defines a problem as a series of limitations on potential solutions, restricting the search space and making problem-solving more efficient.

### ELI5

Imagine we’re playing a game of Sudoku. We know each row, column, and square must have all the numbers from 1 to 9 with no repetition. These are our constraints. Constraint Programming is just like that – we give the computer a problem and define what “rules” or constraints a solution must follow. The computer then finds a solution that satisfies all these rules.

### In-depth explanation

Constraint Programming (CP) is an efficient and powerful paradigm in artificial intelligence (AI) and operations research, generally applied to solve combinatorial problems. The idea behind CP is problem modelling: specific constraints define the relationships between variables, and these relationships must hold true for a solution to be valid.

Representing a problem as a set of constraints simplifies the problem-solving process. It relies on two fundamental components:

- Variables: These are the elements of the problem that we can change and that we are trying to find values for.
- Constraints: These are the rules or limitations that define the relationships between all variables.

Distinct from other techniques like operational research methods or heuristic algorithms, CP provides a higher level of abstraction. This sets CP apart, as it allows for a more intuitive problem representation framework. Solving a constraint satisfaction problem (CSP) means finding a value for each variable that simultaneously satisfies all constraints.

The underlying strength of constraint programming resides in the utilization of specialized algorithms and heuristics to constrain the problem-solving search space. This optimization process, known as ‘constraint propagation’, prunes unlikely solutions, making the problem-solving process faster and more efficient.

One widespread application of CP is in scheduling, such as job sequencing and timetabling problems. Nevertheless, it is sufficiently general to solve a wide range of problems, and other application areas include planning, routing, and configuration.

Furthermore, CP is not restricted to problems where all constraints are known or fixed. For dynamic or uncertain environments, additional techniques like constraint relaxation can be used, resulting in more flexible, robust solutions.

Python, for instance, offers libraries such as Google’s OR-Tools that support constraint programming. example = cp_model.CpModel()

# Variables.

var = [] for j in range(5): var.append(model.NewIntVar(0, 1, str(j)))

# Constraint: at most 2 variables can be true.

model.Add(sum(var) <= 2)

solver = cp_model.CpSolver() result = solver.Solve(model)

### Related terms

Propositional Logic, Satisfiability Problem (SAT), Boolean Satisfiability (SAT) Problem, Mixed Integer Programming (MIP), Heuristics, Operations Research (OR), Combinatorial Optimization, Constraint Satisfaction Problem (CSP), Dynamic Constraint Satisfaction Problem (DCSP)