Approximationdepends

Approximationdepends is a concept in the study of approximation algorithms that describes how the guaranteed quality of a solution depends on specific instance parameters. The term is used to denote that the approximation ratio is not determined solely by the algorithm itself but also by measurable characteristics of the input, such as size, density, weight distribution, or constraint structure.

Formally, for an optimization problem and an algorithm A running in polynomial time, the approximation ratio on an instance I is R_A(I) = OPT(I)/ALG(I). Approximationdepends refers to the property that there exists a function f of instance parameters p(I) such that R_A(I) ≤ f(p(I)) for all I, or equivalently that the worst-case bound varies with p(I). This framing emphasizes how performance varies across different classes of instances rather than across all inputs of the same size.

Typical focus areas include parameterized approximation, where a parameter k influences the bound, and data-dependent guarantees, where distributions of inputs yield tighter ratios. Examples arise in scheduling, facility location, or network design problems where parameters like maximum load, sparsity, or weight skew influence the achievable approximation.

Relation to other concepts: approximationdepends complements parameterized complexity and instance-optimal analyses by highlighting parameter-driven changes in approximation quality. It is used as a lens to identify instance classes where simple algorithms perform well and to guide the development of adaptive strategies.

Criticism and limitations: since f can be problem-specific and not universal, reported bounds may be of limited generality. As such, the term serves more as a descriptive property than a standard metric, and its precise meaning can vary between authors. See also parameterized approximation, instance-optimality, and adaptive algorithms.