Inapproximabilityvoimakkuuksiin

Inapproximabilityvoimakkuuksiin is a term used in computational complexity to describe the study of how the hardness of approximating optimization problems scales when the target approximation factor is allowed to grow as a function of the input size, typically as a power. The idea combines inapproximability results with the analysis of approximation ratios that are polynomially large or small, such as factors of n^α or poly(n). The field builds on gap reductions and the PCP theorem to show that certain problems remain hard to approximate even when the allowed error margin is a polynomial function of the input length.

The standard framework uses reductions that preserve approximation gaps and yields lower bounds of the form that no polynomial-time algorithm can guarantee a solution within a given function f(n) of the optimum unless P = NP. Classic results in this area include strong inapproximability for Maximum Independent Set and, more generally, for many dense combinatorial problems. For example, results attributed to Hastad show that MIS cannot be approximated within n^{1−ε} for any ε > 0 unless P = NP. Set Cover and related covering problems also exhibit hardness that aligns with the best available polylogarithmic or logarithmic approximation algorithms.

Applications and research directions focus on clarifying the limits of fast algorithms, guiding the design of heuristics, and understanding the boundary between approximation schemes (such as PTAS) and inapproximability as the target factor grows with input size. See also inapproximability, hardness of approximation, PCP theorem, NP-hardness, and approximation algorithms.