Is there an efficient algorithm that allows to construct a cryptographically secure $n$-bit hash function based on two unkeyed $n$-bit permutations? Let $f(x)$ and $g(x)$ denote two independent, “ideal”, unkeyed, public $n$-bit permutations (two publicly known bijective functions that produce $n$-bit outputs from $n$-bit inputs), where $n$ is an arbitrary natural number greater than or equal to $3$ (the number $3$ is here because it is the minimal size of a “cryptographically significant” S-Box).  

Does there exist an efficient iterative algorithm (based on $f(x)$ and $g(x)$ as the underlying functions) that allows to construct a cryptographically secure $n$-bit hash function for $k$-bit inputs, assuming that $k$ is an arbitrary natural number greater than or equal to zero (that is, there should be no limit on the maximum length of the input)?
It is impossible to construct a collision-resistant hash function from one-way permutations, via a black-box construction. Thus, the answer in principle is no; such a construction is impossible. Finding collisions on a one-way street: Can secure hash functions be based on general assumptions? by Simon (Eurocrypt'98). This answer does not relate to heuristic constructions, but you cannot do this with a proof of security (unless it's not black box).

The question here actually relates to "ideal" permutations, and under such an assumption it may be different. (Although, I'm not sure what is meant by "public" ideal.) My answer referred to (non-ideal) one-way permutations.