  Even if Supersymmetric particles are found at the Large Hadron Collider
(LHC), it will be difficult to prove that they constitute the bulk of the Dark
Matter (DM) in the Universe using LHC data alone. We study the complementarity
of LHC and DM indirect searches, working out explicitly the reconstruction of
the DM properties for a specific benchmark model in the coannihilation region
of a 24-parameters supersymmetric model. Combining mock high-luminosity LHC
data with present-day null searches for gamma-rays from dwarf galaxies with the
Fermi LAT, we show that current Fermi LAT limits already have the capability of
ruling out a spurious Wino-like solution that would survive using LHC data
only, thus leading to the correct identification of the cosmological solution.
We also demonstrate that upcoming Planck constraints on the reionization
history will have a similar constraining power, and discuss the impact of a
possible detection of gamma-rays from DM annihilation in Draco with a CTA-like
experiment. Our results indicate that indirect searches can be strongly
complementary to the LHC in identifying the DM particles, even when
astrophysical uncertainties are taken into account.
