Reducing problems to solve easier problems

Question

Is there any instance where a problem $A$ can be reduced to a problem $B$ where $B$ is easier to solve than $A$?

I've been learning about NP-Hardness recently and seems that the answer is no. Whenever we show $ A \leq_p B$ its also said that $B$ must therefore be at least as hard as $A$. If this is always the case then is reducing a problem into another only useful for proving NP-Hardness/NP-Completeness? Is there no way to leverage reductions to find an easier problem to solve than the one you originally start out with?

Answer 1

Roughly speaking, no. One way to solve A is by using the method for solving B combined with the reduction from A to B. So if that reduction is efficient, then solving A can't be too much harder than solving B.

If I'm pedantic, yes, B might be a little bit easier to solve than A. If the reduction is very expensive (takes a long time, or significantly increases the size of the problem instance), then B might be a lot easier to solve than A. Normally in study of NP-completeness we address this by requiring the reduction to run in polynomial time, and we ignore any differences that are at most a polynomial factor.