  We discuss a possibility that the parameter space of the two Higgs doublet
model is significantly narrowed down by considering the synergy between direct
searches for additional Higgs bosons at the LHC and its luminosity upgraded
operation and precision measurements of the Higgs boson properties at future
electron-positron colliders such as the International Linear Collider. We show
that, in the case where the coupling constants of the discovered Higgs boson
are slightly different from the predicted values in the standard model, most of
the parameter space is explored by the direct searches of extra Higgs bosons,
in particular for the decays of the extra Higgs bosons into the discovered
Higgs boson, and also by the theoretical arguments such as perturbative
unitarity and vacuum stability. This can be done because there appears an upper
limit on the mass of the extra Higgs bosons as long as the deviation exists in
the Higgs boson coupling. We also show that in the alignment limit where all
the Higgs boson couplings take the standard model like values most of the
parameter space cannot be excluded because most of the Higgs to Higgs decays
are suppressed and also there is no upper limit on the masses from the
theoretical arguments.
