What's the (asymptotic) solution of the recurrence $T(n,m,k,t)\leq T(n^\frac 1 m, 1, km, kt) + \Theta(n)$?
I know how to solve univariate recurrences, but this recurrence is much more difficult, so I am stuck here.
The solution of this inequality should be something that does not imply any "dependency" between $n,m,k,t$. For example, subtituting the solution $T(n,m,k,t)=n^\frac k m$ yields $n^\frac m k\leq n^k$ which is true for every $n,m,k,t\geq 1$, while $T(n,m,k,t)=nk$ implies $nk\leq n^\frac 1 m km$ which is equaivalent to $n\leq n^\frac 1 m m$ which is not always true, and thus implies some dependency between $n,m$. Therefore, this solution holds only in special cases that satisfy this dependency.
I believe that the set of solutions of this inequality is a subset of the set of solutions that @D.W. has shown (as some of which are true only assuming some dependency between $m,n,k,t$).