What does FTP and XP stand for in "$FTP \subsetneq XP$?

Question

What do FPT and XP stand for in this question? Proving FPT is strictly contained in XP

I don't have enough reputation to comment on that post and ask and I couldn't find the meanings of these acryonyms through Google

Answer 1

FPT stands for fixed-parameter tractable, and is the class of parameterized problems solvable in time $$f(k) \cdot n^{O(1)},$$ for any function $f$. A parameterized problem is a problem that comes with a specific parameter that you may use exponential time in, or that we are particularly interested in. Examples of parameterized problems are Vertex Cover parameterized by solution size, Independent Set parameterized by solution size, coloring parameterized by number of colors, etc. Only the first of these belong to FPT, where as the second belong to a class W[1] (if you're going to ask, W is short for weft, don't ask).

There are few problems that make sense to cite as being in XP, (the third problem is, as Juho correctly points out para-NP-hard and not in XP). XP is "short" for slice-wise polynomial, and is the class of problems solvable in time $O(n^{f(k)})$. Note that coloring cannot be in XP (unless P = NP) since then 3-coloring (which is NP-complete) would be solvable in $O(n^{f(3)}) = O(n^c)$ time, thus rendering it polynomially solvable.

Answer 2

From a quick google search, I found out this Wikipedia page on FPT and this site on XP.

If you want to know what I searched: I literally googled "FPT complexity class" and it was the first result :)

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And no, I'm not familiar with those complexity classes as well