Minireductions

A minireduction is a specific form of problem reduction used in theoretical computer science and mathematical logic. It is a transformation that reduces the size of an instance of a decision problem while preserving its solvability, typically by removing redundant structure or simplifying the representation. The term emphasizes the reduction’s compactness; the resulting instance is usually only marginally larger in terms of input length compared to the original.

The concept was introduced in the early 1990s as a refinement of the standard polynomial-time reduction. Researchers sought to produce more efficient reducibility proofs for NP‑complete problems, and the minireduction framework allowed them to construct smaller hard instances that maintained the essential complexity characteristics. Because the transformation is lightweight, proofs that rely on minireductions can often be verified more quickly by computer-assisted proof assistants.

Minireductions are employed in a variety of contexts. In graph theory they can reduce a large graph to a smaller core while preserving properties such as colorability or planarity. In satisfiability research they are used to transform a formula into a minimized equivalent that still captures the hardness of the original problem. They also appear in kernelization studies where large problem instances are pre‑processed into smaller “kernels” that are easier to solve.

A typical minireduction step might delete isolated vertices, merge duplicate constraints, or convert a three‑literal clause into a two‑literal clause by introducing a new auxiliary variable that preserves the satisfiability status. Because the resulting instance is only a constant factor larger, the reduction is considered “minimally invasive.”

Minireductions contribute to the understanding of computational complexity classes, particularly in the design of tight lower bounds and in the analysis of approximation algorithms. They are routinely cited in research on graph minors, SAT‑solving heuristics, and formal verification.