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Clash Royale CLAN TAG #URR8PPP up vote 3 down vote favorite I read that a fully polynomial time approximation scheme (FPTAS) has time complexity polynomial in the input size and also polynomial in $1/epsilon$. However, this leaves some ambiguity. For example, $T(n,1/epsilon)=min(n^1/epsilon,(1/epsilon)^n)$ is both polynomial in $n$ (when holding $1/epsilon$ constant) and polynomial in $1/epsilon$ (when holding $n$ constant). But $T$ is not polynomial in $n/epsilon$ or $(n+1/epsilon)$ because $sqrt x^sqrt x$ and $(x/2)^x/2$ aren't bounded by polynomials. Does this mean an algorithm with running time $T$ isn't an FTPAS, and perhaps a better definition would be that it has to be polynomial in $n+1/epsilon$? algorithms complexity-theory terminology share | cite | improve this question asked 2 days ago Akababa 118 4 add a comment  |  up vote 3 down vote favorite I read that a fully polynomial time approximation...

Clash Royale CLAN TAG #URR8PPP up vote 16 down vote favorite 2 Consider the birthday problem. Given $N$ people, how many ways are there for there to exist some pair of people with the same birthday? Enumerating the possibilities quickly becomes tedious However, the complement problem (Given $N$ people, how many ways are there for no one to have the same birthday?) is trivial. In fields like probability, this has obvious applications, due to the "complement law": if $A cup A^c = S$, where $S$ is the entire sample space, then $$P(A) + P(A^c) = 1 implies P(A) = 1 - P(A^c)$$ In general, this pattern is very common. Intuitively, I sense: somehow, the complement problem is asking for a lot less information if one has something like the "complement law" in probability, then in some restricted scope of problems, the "complement law" gives in some sense, the "same amount of information" What do mathematicians call what I am getting at here? A...