Subreductions

A subreduction is a concept encountered in certain areas of mathematics and computer science, particularly in the study of algorithms and complexity theory. It refers to a transformation or mapping of one problem, P1, into another problem, P2, such that a solution to P2 can be used to efficiently construct a solution to P1. This transformation must be computable in polynomial time, meaning the time it takes to perform the transformation grows no faster than a polynomial function of the input size.

The primary purpose of a subreduction is to demonstrate relationships between problems. If problem P1 can be subreduced to problem P2, it implies that P1 is "no harder" than P2 in terms of computational resources. Specifically, if P2 can be solved efficiently, then P1 can also be solved efficiently by first transforming P1 into P2 and then solving P2. This is a key technique used in proving the hardness of problems. For instance, if a known hard problem P_hard can be subreduced to a new problem P_new, then P_new must also be at least as hard as P_hard, suggesting that P_new might also be computationally difficult.

Subreductions are fundamental to the theory of NP-completeness. A problem is NP-complete if it is in the complexity class NP and every other problem in NP can be reduced to it. The "reduction" commonly referred to in this context is often a form of subreduction, where the transformation is efficient. This concept helps categorize problems based on their inherent difficulty and understand the landscape of computational complexity.