# What does the complexity class $\mathsf{XP}$ stand for?

$\mathsf{XP}$ is the class of problems with input length $n$ and parameter $k$ than can be solved in $O(n^{f(k)})$ time, where $f$ is a computable function. It's described on the complexity zoo page as "Fixed-parameter Tractable for Each Parameter", but I fail to see how adding "for each parameter" to the description causes such a significant change to the definition compared to $\mathsf{FPT}$, which contains problems solvable in $O(f(k)\cdot n^{O(1)})$ time.

The complexity zoo page mentions the book it was defined in, Parameterized Complexity, but I don't have access to it right now.

• It seems to me that the essential part of the definition is missing. FPT says for some $k$ and it sounds like XP would like to say something for all $k$. – Raphael Jun 9 '17 at 4:50
• XP stands for "slice-wise polynomial". – Juho Jun 9 '17 at 5:30

There is a slightly different description for XP which I personally find less misleading: "Polynomial time for each parameter". I believe the zoo page uses FPT instead of P for some formal reasons (parameterized problems are sometimes defined as subsets of $\Sigma^*\times\Pi^*$, classical problems as subsets of $\Sigma^*$). It can be noted that both formulations describe non-uniform XP, while the time bound $O(n^{f(k)})$ refers to uniform XP. In any case, the important difference is that for XP, the exponent of $n$ may depend on the parameter, for FPT it mustn't.