Explain the concept of polynomial-time reduction.

Polynomial-time reduction is a concept in computational theory that involves transforming one computational problem into another in a way that preserves the complexity of the problem. Specifically, it aims to show that if problem A can be solved in polynomial time, then problem B can also be solved in polynomial time.

In a polynomial-time reduction, an algorithm is designed to convert instances of problem A into instances of problem B. This conversion should be done in polynomial time, meaning that the time required to perform the conversion is bounded by a polynomial function of the input size.

Furthermore, the reduction should ensure that the solution to problem B can be used to solve problem A. This means that if we have an algorithm that can solve problem B in polynomial time, we can use it to solve problem A by applying the reduction algorithm to convert the instance of problem A into an instance of problem B, solving it using the algorithm for problem B, and then converting the solution back to a solution for problem A.

By demonstrating a polynomial-time reduction from problem A to problem B, we establish a relationship between the complexities of the two problems. If problem A is known to be NP-hard (not solvable in polynomial time), and we can show a polynomial-time reduction from problem A to problem B, then problem B is also NP-hard. Conversely, if problem B is known to be solvable in polynomial time, and we can show a polynomial-time reduction from problem A to problem B, then problem A is also solvable in polynomial time.