  We consider the dynamics of $N$ interacting bosons initially exhibiting
Bose-Einstein condensation. Due to an external trapping potential, the bosons
are strongly confined in two spatial directions, with the transverse extension
of the trap being of order $\varepsilon$. The non-negative interaction
potential is scaled such that its scattering length is positive and of order
$(N/\varepsilon^2)^{-1}$, the range of the interaction scales as
$(N/\varepsilon^2)^{-\beta}$ for $\beta\in(0,1)$. We prove that in the
simultaneous limit $N\rightarrow\infty$ and $\varepsilon\rightarrow 0$, the
condensation is preserved by the dynamics and the time evolution is
asymptotically described by a cubic defocusing nonlinear Schr\"odinger equation
in one dimension, where the strength of the nonlinearity depends on the
interaction and on the confining potential. This is the first derivation of a
lower-dimensional effective evolution equation for singular potentials scaling
with $\beta\geq\frac12$ and lays the foundations for the derivation of the
physically relevant one-dimensional Gross-Pitaevskii equation ($\beta=1$). For
our analysis, we adapt an approach by Pickl to the problem with strong
confinement.
