This is a continuation of my attempts to generate simple combinatorial computational problems that turn out to be computationally hard (NP-complete). In this pursuit, I came up with a permutation problem (How hard is reconstructing a permutation from its differences sequence?) which turned out to be NP-complete. Here, I propose another computational problem which I highly suspect that it is NP-complete.
The shift of a permutation $\pi(i)$ of numbers $1, 2, \ldots n$ is defined as $\sigma(i)$= $\pi(i+k \mod n)$ for some fixed $k$, $1 \le k \lt n$. The shift product of a permutation $\pi$ is defined as $ \pi(i+k \mod n) \circ \pi(i)$. We say that a permutation $\pi$ is a square for the shift product if there is a permutation $\tau$ such that $\pi$ is the shift product of $\tau$ (i.e. $\pi(i)= \tau(i+k \mod n) \circ \tau(i)$).
I did not find a notion of shift product for permutations in the literature.
Instance: A permutation $\pi$ of $1, 2, \ldots n$
Question: Is permutation $\pi$ a square for the shift product?
Is there an efficient algorithm to determine whether a given permutation $\pi$ is a square for the shift product of some permutation $\tau$, or is it NP-complete?
The problem appears to be hard even when the shift amount $k=1$.
This was posted on mathOverflow.