Say we have a computational problem, e.g. 3-SAT, that has a set of problem instances (possible inputs) $S$.
Normally in the analysis of algorithms or computational complexity theory, we have some sets
$$I(n) = {w in S : |w| = n}$$
of all inputs of length $n$, and a function $T(w)$ that gives the running time of some solution algorithm $A$ on input $w$.
The worst-case running time sequence for $A$ is then
$$ f_n = max_{w in I(n)} T(w). $$
Let us now define the sets
$$ I^K(n) = {w in S : K(w) = n }$$
of all inputs with Kolmogorov complexity $n$, and let us define the sequence
$$ f^K_n = frac{1}{left|I^K(n)right|} sum_{w in I^K(n)} T(w). $$
Here $f^K$ is the average running time sequence for $A$, except where the “size” of inputs is their Kolmogorov complexity, not their length.
Are there algorithms for which $f_n$ is asymptotically significantly different from $f^K_n$? If so, are there problems whose time complexity changes when using this different way of analyzing algorithms?