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Why are polynomial-time algorithms usually considered efficient?

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A three-dimensional Lorenz attractor graph with swirling blue lines forming a butterfly-shaped pattern, representing chaotic motion in a mathematical weather model, with labeled x, y, and z axes. — complexity The weather model of meteorologist Edward Lorenz
The x-, y-, and z-axes correspond to components of the weather, while the graph indicates feasible weather patterns. The graph is a function of time, which means, in theory, that tracing the curve in one direction reveals past weather patterns, while tracing the curve in the other direction predicts future weather patterns. However, practically indistinguishable points may lead to completely different weather epochs, indicated by the two distinct “lobes.” This phenomenon, known as sensitivity to initial conditions, is what prevents precise weather forecasts more than a few days into the future.

Polynomial-time algorithms are considered efficient because the number of computational steps required to solve a problem grows at a manageable rate as the input size ( $n$ ) increases. Specifically, the execution time for these algorithms is bounded by a polynomial function of $n$ .123

In contrast, algorithms that require exponential time are considered inefficient or intractable because their execution times grow much more rapidly as the problem size increases. For instance, an algorithm with exponential time complexity, such as $O (2^{n})$ , can quickly become impractical; while it might solve a problem for a small $n$ in less than a second, the same algorithm could take tens of thousands of years to complete as $n$ grows to 100.13

Because polynomial-time algorithms avoid this explosive growth in required operations, they are classified as tractable and are the standard for efficient computation in computer science.12