combinationto

Combinationto is a term used in combinatorial optimization and constraint programming to denote an operation that merges two or more constraint systems or decision components into a single, integrated structure. In its simplest form, a combinationto takes two constraint networks CN1 = (V1, D1, C1) and CN2 = (V2, D2, C2) and produces CN = CN1 ⊗ CN2 with V = V1 ∪ V2, D assigning domains to variables, and C including C1 ∪ C2 plus cross-constraints that specify how variables in V1 relate to variables in V2. The cross-constraints are defined by a relation R ⊆ Dom1 × Dom2, and a feasible assignment is one that satisfies all local constraints and all cross-constraints.

Properties and considerations: The outcome depends on the cross-constraint relation R; the operation is generally not commutative or associative without additional structure. It is often monotone with respect to domain refinement, and can be used within search procedures that propagate constraints jointly across subsystems.

Examples: Suppose CN1 constrains a variable x in {0,1} and CN2 constrains y in {0,1}, with a cross-constraint R enforcing x ≥ y. The combinationto CN enforces both local constraints and the cross-constraint, yielding feasible pairs (x,y) such as (1,0) and (1,1).

Origins and usage: The term appears in recent theoretical discussions and workshop notes as a concise label for modular combination of constraint systems. It is used to model interactions between components in modular design, scheduling with cross-system dependencies, and configuration problems where parts must be compatible.

See also: product of constraint networks, constraint propagation, modular design, combinatorial optimization, constraint satisfaction problem.