  We investigate the replica theory of the liquid-glass transition for a binary
mixture of large and small additive hard spheres. We consider two different
ans\"atze for this problem: the frozen glass ansatz (FGA) in whichs the
exchange of large and small particles in a glass state is prohibited, and the
exchange glass ansatz (EGA), in which it is allowed. We calculate the dynamical
and thermodynamical glass transition points with the two ans\"atze. We show
that the dynamical transition density of the FGA is lower than that of the EGA,
while the thermodynamical transition density of the FGA is higher than that of
the EGA. We discuss the algorithmic implications of these results for the
density-dependence of the relaxation time of supercooled liquids. We
particularly emphasize the difference between the standard Monte Carlo and swap
Monte Carlo algorithms. Furthermore, we discuss the importance of particle
exchange for estimating the configurational entropy.

Soft Condensed Matter;Statistical Mechanics