  In this paper, we propose a subnetwork decomposition/combination approach to
investigate the single rate $2$-pair unicast problem. It is shown that the
solvability of a $2$-pair unicast problem is completely determined by four
specific link subsets, namely, $\mathcal A_{1,1}$, $\mathcal A_{2,2}$,
$\mathcal A_{1,2}$ and $\mathcal A_{2,1}$ of its underlying network. As a
result, an efficient cut-based algorithm to determine the solvability of a
$2$-pair unicast problem is presented.
