  The main goal of the experimental programs on ultra-relativistic heavy ion
collisions at the LHC is the production and characterization of the Quark Gluon
Plasma (QGP), a phase of nuclear matter in which strongly interacting
constituents (quarks and gluons) are deconfined. Heavy quarks are considered
effective probes of the properties of the QGP as they are created on a short
time scale, with respect to that of the QGP, and subsequently interact with it.
Moreover, for a proper assessment of the characteristics of the matter produced
in heavy-ion collisions, it is important to disentangle the final state effects
due to the formation of a QGP from the initial state effects due to the fact
that nuclei are present in the colliding system. Both initial and final state
effects may lead to qualitatively similar phenomena in the observables of
interest. The measurement of charmed meson production in proton-nucleus
collisions allows to assess initial state effects present in nuclear
collisions, under the assumption that an extended deconfined medium is not
created in this kind of interactions. The nuclear modification factor of $D$
mesons in p-Pb collisions ($R_{\rm pPb}$) is essential for a complete
understanding of the modification of $D$ mesons momentum distributions observed
in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$= 2.76 TeV, which is interpreted as
due to the $c$-quark energy loss in the medium. In addition, some of the
results obtained from high-multiplicity p-Pb collisions at LHC, such as the
ridge structure in the two-particle correlation function, turned out to be
unexpected, and have been interpreted in terms of final state effects such as
hydrodynamic flow. These aspects make a study of charmed meson production in
p-Pb collisions as a function of the event multiplicity of great interest.
