  Recently, Hajdu and S\'{a}rk\"{o}zy studied the multiplicative decompositions
of polynomial sequences. In particular, they showed that when $k \geq 3$, each
infinite subset of $\{x^k+1: x \in \mathbb{N}\}$ is multiplicatively
irreducible. In this paper, we attempt to make their result effective by
building a connection between this problem and the bipartite generalization of
the well-studied Diophantine tuples. More precisely, given an integer $k \geq
3$ and a nonzero integer $n$, we call a pair of subsets of positive integers
$(A,B)$ \emph{a bipartite Diophantine tuple with property $BD_k(n)$} if
$|A|,|B| \geq 2$ and $AB+n \subset \{x^k: x \in \mathbb{N}\}$. We show that
$\min \{|A|, |B|\} \ll \log |n|$, extending a celebrated work of Bugeaud and
Dujella (where they considered the case $n=1$). We also provide an upper bound
on $|A||B|$ in terms of $n$ and $k$ under the assumption $\min \{|A|,|B|\}\geq
4$ and $k \geq 6$. Specializing our techniques to Diophantine tuples, we
significantly improve several results by B\'{e}rczes-Dujella-Hajdu-Luca,
Bhattacharjee-Dixit-Saikia, and Dixit-Kim-Murty.
