The given reaction is an $S_N2$ nucleophilic substitution reaction, as it involves a primary alkyl halide (1-bromopropane) reacting with a strong nucleophile ($\text{CN}^-$). $S_N2$ reactions proceed fastest in polar aprotic solvents. These solvents effectively solvate cations (like $\text{Na}^+$) but do not strongly solvate anions (like $\text{CN}^-$) via hydrogen bonding, leaving the nucleophile highly reactive. Among the given options, water, methanol, and ethanol are polar protic solvents. N,N'-dimethylformamide (DMF) is a polar aprotic solvent, making it the ideal choice to accelerate this $S_N2$ reaction.