  Cold Dark Matter theory predicts that the Local Group should contain many
more dwarf-sized objects than the observed number of dwarf galaxies --- the
so-called sub-structure problem. We investigate whether the suppression of star
formation in these small objects due to the presence of a photoionizing
background can resolve the problem. We make use of results from recent
hydrodynamic simulations to build a recipe for the suppression of gas infall
into semi-analytic galaxy formation models, and use these to predict the
luminosity function of dwarf galaxies in the Local Group. In the models without
photoionization ``squelching'', we predict a large excess of faint dwarf
galaxies compared with the observed number in the Local Group --- thus, the
usual recipe for supernovae feedback used in semi-analytic models does not
solve the sub-structure problem on its own. When we include photoionization
squelching, we find good agreement with the observations. We have neglected
tidal destruction, which probably further reduces the number of dwarf galaxies.
We conclude that photoionizing squelching easily solves the sub-structure
problem. In fact, it is likely that once this effect is taken into account,
models with reduced small-scale power (e.g. Warm Dark Matter) would
underproduce dwarf galaxies.
