Mapping supernovae to their progenitors is fundamental to understanding the collapse of massive stars. We investigate the red supergiant problem, which concerns why red supergiants with masses $sim16$-$30 M_odot$ have not been identified as progenitors of Type IIP supernovae, and the supernova rate problem, which concerns why the observed cosmic supernova rate is smaller than the observed cosmic star formation rate. We find key physics to solving these in the compactness parameter, which characterizes the density structure of the progenitor. If massive stars with compactness above $xi_{2.5} sim 0.2$ fail to produce canonical supernovae, (i) stars in the mass range $16$-$30 M_odot$ populate an island of stars that have high $xi_{2.5}$ and do not produce canonical supernovae, and (ii) the fraction of such stars is consistent with the missing fraction of supernovae relative to star formation. We support this scenario with a series of two- and three-dimensional radiation hydrodynamics core-collapse simulations. Using more than 300 progenitors covering initial masses $10.8$-$75 M_odot$ and three initial metallicities, we show that high compactness is conducive to failed explosions. We then argue that a critical compactness of $sim 0.2$ as the divide between successful and failed explosions is consistent with state-of-the-art three-dimensional core-collapse simulations. Our study implies that numerical simulations of core collapse need not produce robust explosions in a significant fraction of compact massive star initial conditions.
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