  Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors,
counted according to multiplicity. Suppose that $a$ and $q$ are positive
integers satisfying $(a,q)=1$. Denote by $\mathcal{P}_2(a,q)$ the least
almost-prime $\mathcal{P}_2$ which satisfies $\mathcal{P}_2\equiv a\pmod q$. In
this paper, it is proved that for sufficiently large $q$, there holds
\begin{equation*}
  \mathcal{P}_2(a,q)\ll q^{1.82193}. \end{equation*} This result constitutes an
improvement upon that of Iwaniec, who obtained the same conclusion, but for the
range $1.845$ in place of $1.82193$.
