Divisiondepends

Divisiondepends is a theoretical concept used to describe how the outcome of a computation, optimization, or decision problem can depend on the particular way a problem is divided into subproblems. Unlike division-agnostic formulations, where subproblem solutions can be combined to produce a global solution independent of the partition, divisiondepends captures dependence of subproblem results on the chosen partitioning scheme.

In formal terms, consider a problem P that can be decomposed into subproblems P1,...,Pk. If there exists a partition function Par that determines a division of P into these subproblems such that the computed subproblem solutions S1,...,Sk, and hence the overall solution S, are not guaranteed to be determined by P alone but also by Par, then P exhibits divisiondepends.

Applications and implications: The concept highlights potential pitfalls in divide-and-conquer strategies, dynamic programming with data-dependent decomposition, and distributed computation where partitioning affects results or performance. It is relevant when subproblem interactions are non-local or when combining rules introduce nonlinearity.

For instance, in a scheduling problem where task groups are formed based on current workload versus geographic constraints, different groupings yield different total completion times, demonstrating divisiondepends.

The term is used informally in some theoretical discussions to illustrate the sensitivity of problems to decomposition choices. It is not widely standardized and should be defined in context when used.

See also: divide-and-conquer, problem decomposition, partitioning, dependency graph.