Call a function$$\mathbb{N}\times \mathbb{N}\to \{0, 1\}, \quad (n, m)\to f(n, m)$$computable in polynomial time in $\log n+\log m$ a PTIME family.


Given a PTIME family $f$ call a computable function $g:\mathbb{N}\to \mathbb{N}$ such that $f(n, g(n))=1$ for all $n\in \mathbb{N}$ a solution of $f$.


Is there a PTIME family $f$ that has at least one solution and such that for each solution $g$ and each polynomial $P$ there exists $n\in \mathbb{N}$ such that the running time of $g$ on $n$ is more than $P(\log n+\log g(n))$?


The reason for taking $\log n+\log g(n)$ is to ensure that neither input nor output dominates the running time of the algorithm.


$\log 0=0$ by convention.



Family of PTIME sets where it is hard to name elements
This is an open problem.


If $\mathrm{TFNP\ne FP}$, then a TFNP problem outside FP directly gives what you call a PTIME family (with solutions polynomially bounded in terms of the input) whose solutions cannot be computed in polynomial time.


If P = NP, then no such problem exists: the assumption implies that given $n$ and $m$, we can compute in polynomial (in $\log n$ and $\log m$) time a solution $y$ such that $f(n,y)=1$ and $y\le m$ if it exists (see e.g. Arora&Barak, Theorem 2.18). Let’s call this algorithm $h(n,m)$. Then successively call $h(n,1)$, $h(n,2)$, $h(n,4)$, $h(n,8)$, ... until you find a solution; this will take time polynomial in $\log n$ and $\log y$ where $y$ is the least solution, because the last call of $h(n,m)$ will be made with $m\le 2y$.


