Complexity Class of an Algorithm with two Inputs

Question

Consider a problem with two inputs like (P,L) and |P|=n and L is some positive integer. If my algorithm had a complexity of O(n^L), would that still be polynomial? Or is it exponential? I'm not sure if I should view L as a constant or as 'size of input', since it's not a size but just some number (as in, Knapsack for example, select items of weight <= L). At the same time, L is part of the input and not fixed.

Answer 1

You're confused because you've called part of your input $n$. When we say that the running time is some function of $n$, we almost always mean that $n$ is the length of the input string.

Your algorithm runs in time $|P|^L$, where $P$ is a string contained in the input and $L$ is a number represented in the input, presumably in binary. If $L$ is a $b$-bit number, it could be as big as $2^b$. Writing $n$ for the length of the input, we could, for example, have $|P|=n/2$ with $L$ being an $(n/2)$-bit number. That case gives running time $(n/2)^{2^{n/2}}$, which is a long, long way from being polynomial.