We present results from the EDGE survey, a spatially resolved CO(1‑0) follow-up to CALIFA, an optical Integral Field Unit survey of local galaxies. By combining the data products of EDGE and CALIFA, we study the variation in molecular gas depletion time (τ dep) on kiloparsec scales in 52 galaxies. We divide each galaxy into two parts: the center, defined as the region within 0.1 {R}25, and the disk, defined as the region between 0.1 and 0.7 {R}25. We find that 14 galaxies show a shorter τ dep (∼1 Gyr) in the center relative to that in the disk (τ dep ∼ 2.4 Gyr), which means the central region in those galaxies is more efficient at forming stars per unit molecular gas mass. This finding implies that the centers with shorter τ dep resemble the intermediate regime between galactic disks and starburst galaxies. Furthermore, the central drop in τ dep is correlated with a central increase in the stellar surface density, suggesting that a shorter τ dep is associated with molecular gas compression by the stellar gravitational potential. We argue that varying the CO-to-H2 conversion factor only exaggerates the central drop of τ dep.