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Is there any tool for this task ?

Given the asymptotic bound in term of $n$ and other paramaters $t_1,\dots,t_r$, then return the value for each $t_i$ which optimizes the expression in term of $n$, asymptotically (may be upto constant factor $O(1)$, or even polynomial factor $n^{O(1)}$.

For example,

  • $O(n/t+t)$, then $t=\sqrt{n}$ gives us O($\sqrt{n}$).
  • $O(n/t+2^t)$, then $t=\log{n}-\log\log{n}$ gives us O($n/\log{n}$).

These are quite simple.

But when I see something like this from the paper..

  • $\Omega( (\binom{n+k}{2k} \binom{n^2+l-2k}{l})/ (\binom{\sqrt{n}+k}{k}\binom{n^2+l+\sqrt{n}k}{n^2})$. Then $l=n^2t$ and $k=\Theta(n/t)$ gives us $\exp(\Omega(\sqrt{n}))$.

I hope there are the computer tools which can optimize this.

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