Constraint Programming (CP)
Constraint Programming (CP) is a powerful paradigm for modeling and solving combinatorial decision problems. Unlike mathematical programming approaches such as Linear Programming (LP), Mixed-Integer Linear Programming (MILP), or Mixed-Integer Nonlinear Programming (MINLP), constraint programming focuses on expressing logical relationships and combinatorial structure directly, rather than optimizing a numerical objective under algebraic constraints.
CP is widely used in scheduling, timetabling, resource allocation, configuration, and planning, where feasibility, logical consistency, and complex constraints are often more important than linear or convex structure.
What is Constraint Programming (CP)?
In Constraint Programming (CP), a problem is defined by:
- A set of decision variables, each with a finite domain of possible values,
- A set of constraints describing which combinations of values are allowed,
- Optionally, an objective function to optimize.
Constraints can be highly expressive and non-algebraic, such as:
- All-different constraints,
- Logical implications,
- Temporal and precedence constraints,
- Global constraints capturing complex patterns (e.g., cumulative, element, or regular constraints).
The emphasis is on feasibility and constraint satisfaction, though optimization variants exist and are commonly used.
Constraint propagation and search
The core mechanism behind Constraint Programming (CP) is constraint propagation. When a variable’s domain is reduced, constraints infer and remove inconsistent values from other variables’ domains. This pruning reduces the search space dramatically before and during search.
Once propagation alone cannot determine a solution, Constraint Programming (CP) solvers employ backtracking search, systematically exploring variable assignments. Modern solvers enhance this with:
- Domain splitting and branching heuristics,
- Conflict-directed backtracking,
- Nogood learning and techniques inspired by SAT solving, such as Conflict-Driven Clause Learning (CDCL).
These techniques differ markedly from Branch-and-Bound or Cutting-Plane methods used in MILP, making Constraint Programming (CP) complementary rather than competing in many applications. Indeed, Constraint Programming (CP) is commonly applied to combinatorial problems characterized by discrete variables and complex logical constraints. By relying on constraint propagation and backtracking search rather than mathematical relaxations, it offers an alternative modeling and solution paradigm that complements linear, nonlinear, and mixed-integer optimization techniques.
Relationship to other problem classes
Constraint Programming (CP) differs fundamentally from mathematical programming:
- It does not rely on relaxations, duality, or convexity.
- It naturally models discrete domains and logical constraints.
- It is closely related to SAT and SMT solving, sharing techniques such as clause learning and backtracking.
However, Constraint Programming (CP) can be combined with other optimization paradigms:
- Hybrid CP/MILP approaches integrate constraint propagation with linear relaxations and cutting planes.
- Matheuristics use CP to generate high-quality feasible solutions for MILP or MINLP models.
- CP can serve as a feasibility engine within larger decomposition or hybrid frameworks.
Further reading and references
Montanari, U. (1974). Networks of Constraints: Fundamental Properties and Applications to Picture Processing. Information Sciences, 7(2), 95–132.
One of the earliest papers formalizing constraint networks and constraint satisfaction problems, laying the theoretical foundations for constraint programming.
Rossi, F., van Beek, P., & Walsh, T. (2006). Handbook of Constraint Programming. Elsevier.
A foundational reference volume covering modeling, algorithms, global constraints, and applications of constraint programming across disciplines.
Mackworth, A. K. (1977). Consistency in Networks of Relations. Artificial Intelligence, 8(1), 99–118.
Introduced key consistency concepts (such as arc consistency) that remain central to constraint propagation and CP solvers.