constraintsolving

Constraint solving is a field in computer science focused on finding assignments to variables that satisfy a set of constraints. A constraint-satisfaction problem (CSP) consists of variables, each with a domain of possible values, and constraints that specify allowable combinations of values. A solution assigns to every variable a value from its domain such that all constraints hold. CSPs model a wide range of problems, including scheduling, timetabling, resource allocation, and configuration.

Solving techniques combine constraint propagation and search. Constraint propagation removes values from variable domains that cannot participate in any solution, using notions of local consistency such as arc consistency (AC-3) and stronger forms. During search, assignments are made and propagated; when conflicts arise, backtracking occurs. MAC (maintaining arc consistency) interleaves propagation with search. Global constraints capture common patterns (for example allDifferent, which enforces pairwise distinct values) and are often implemented with specialized propagators for efficiency. Many CSPs are also encoded into SAT or SMT problems or solved by hybrid methods that mix different solving paradigms.

Problems can be solved for existence of a solution, for a complete set of solutions, or for optimization over solutions (e.g., minimize cost or maximize quality) under the constraints. The general problem is NP-complete, but modern constraint solvers employ powerful heuristics and problem-specific propagators to solve large, real-world instances efficiently.

Languages and tools in constraint solving include MiniZinc and Essence for modeling, and solvers such as Choco, Gecode, IBM CP Optimizer, and Google OR-Tools; some systems integrate CSP techniques with SAT/SMT solvers. Constraint solving thus provides a flexible, declarative framework for a broad class of combinatorial problems.