Polynomial-time reductions are a method of transforming one problem into another in a way that ensures if the second problem can be solved quickly (in polynomial time), then the first problem can also be solved quickly. This concept is crucial in understanding the relationships between problems, particularly in complexity theory, where it helps classify problems based on their solvability and computational difficulty. They provide a foundation for proving the hardness of problems by showing that if one hard problem can be transformed into another efficiently, then solving either problem remains equally challenging.
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