Is 3^{o(n)} = 2^{o(n)}?
In the proof which shows that there is no f(k)n^{o(k)} time algorithm for the CLIQUE problem
by reduction from the 3-Coloring problem,
there was a statement saying that f(k)(3^{n/k})^{o(k)} = 2^{o(k)}.
I wonder how it can be shown.
Is there any formal or informal proof?
Answers
The statement you mentioned, f(k)(3^{n/k})^{o(k)} = 2^{o(k)}, appears in the context of complexity theory, particularly in reductions between different computational problems.
In this case, the reduction is used to show that there is no f(k)n^{o(k)} time algorithm for the CLIQUE problem assuming the Exponential Time Hypothesis (ETH), which conjectures that 3-SAT cannot be solved in sub-exponential time. The proof involves reducing the 3-Coloring problem to CLIQUE, thereby establishing a connection between the time complexity of solving CLIQUE and the hardness of 3-Coloring.
Regarding the specific statement f(k)(3^{n/k})^{o(k)} = 2^{o(k)}, let's break it down:
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f(k) is a polynomial function of the parameter k. This reflects the fact that reductions between computational problems often involve polynomial-time transformations.
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3^{n/k} represents the number of possible colorings for a 3-colorable graph of size n, when each vertex is assigned one of k colors. This is related to the 3-Coloring problem.
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o(k) represents a function that grows slower than any polynomial in k. This reflects the idea that the parameter k, which is related to the size of the color classes in the 3-Coloring problem, can vary in a way that makes the problem harder to solve.
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2^{o(k)} represents the running time of an algorithm solving the 3-Coloring problem. It grows exponentially with a function o(k) in the exponent, indicating that the problem becomes harder as k increases.
To prove the equality f(k)(3^{n/k})^{o(k)} = 2^{o(k)}, one would typically employ techniques from complexity theory, such as reductions and asymptotic analysis. The specific details of such a proof would depend on the context and the specific reduction being used.
While it may not be a formal proof, this equation encapsulates the intuition behind the reduction: solving CLIQUE becomes increasingly difficult as the size of the color classes in the corresponding 3-Coloring instance grows, and this difficulty is captured by the exponential growth in the running time of the algorithm.
