Problemraums

Problemraums is a term used in computational problem theory and related fields to denote the collective space of problem instances that belong to a particular class, together with their descriptive parameters and constraints. The concept focuses on the domain of problems a solver may encounter, rather than on any single instance or its solution.

A Problemraum is typically characterized by its instance space, the evaluation or cost structure, and the rules that bound and relate instances. Individual problems within the space are described by features such as size, density, and constraint composition. The boundaries of the space are defined by the types of transformations considered permissible, such as pruning, renaming, or relaxing constraints, which connect related instances or generate new ones.

In practice, Problemraums are used to categorize, compare, and generate problem instances for benchmarking. They enable researchers to study instance hardness, average-case behavior, and the transferability of algorithms across related problems. They are distinct from the solution space or the search space, which concern the set of possible solutions to a given instance, while the Problemraum concerns the domain of problems themselves.

Examples can be found in SAT research (the space of CNF formulas with bounded variables and clauses), combinatorial optimization (routing or scheduling instances with certain graph properties), and planning (initial states, goals, and actions defining a planning problem). Operators such as reductions, relaxations, and compositions allow navigation within a Problemraum or between problem raums, supporting benchmarking and theoretical analysis.

See also: problem space, search space, instance space, benchmark, instance generation.