A note on the parameter $\mathrm{\ell }$ in Buchbinder--Feldman's deterministic submodular matroid algorithm

Buchbinder and Feldman recently gave a deterministic $(1-1\mathrm{/}e-\epsilon )$-approximation for maximizing a non-negative monotone submodular function subject to a matroid constraint, with query complexity ${\tilde{O}}_{\epsilon }(nr)$. Their algorithm uses an integer parameter $\mathrm{\ell }$, which Buchbinder and Feldman fix to $\mathrm{\ell }=1+\lceil 1\mathrm{/}\epsilon \rceil$ via a loose bound on $(1+1\mathrm{/}\mathrm{\ell }{)}^{-\mathrm{\ell }}$. We point out two purely elementary refinements. First, the classical Pólya–Szegő inequality $(1+1\mathrm{/}\mathrm{\ell }{)}^{-\mathrm{\ell }}\le e^{-1}(1+1\mathrm{/}(2\mathrm{\ell }))$ replaces the loose step in their proof and permits $\mathrm{\ell }=\lceil 1\mathrm{/}(2e\epsilon )\rceil$, shrinking the hidden constant in ${\tilde{O}}_{\epsilon }(nr)$ by a factor $\approx 2^{0.816\mathrm{/}\epsilon }$. Second, an alternating-series tail bound for $\log (1+t)$ yields the asymptotically sharp inequality $(1+1\mathrm{/}\mathrm{\ell }{)}^{-\mathrm{\ell }}\le e^{-1}\exp (1\mathrm{/}(2\mathrm{\ell })-1\mathrm{/}(3{\mathrm{\ell }}^2)+1\mathrm{/}(4{\mathrm{\ell }}^3))$, matching the true expansion of $(1+1\mathrm{/}\mathrm{\ell }{)}^{-\mathrm{\ell }}$ through order ${\mathrm{\ell }}^{-3}$ and translating into ${\mathrm{\ell }}_{\star }=1\mathrm{/}(2e\epsilon )-5\mathrm{/}12+O(\epsilon )$. The asymptotic class ${\tilde{O}}_{\epsilon }(nr)$ of the query complexity is unchanged in either case; only the implicit constant in $\epsilon$ is improved. All inequalities in this note are formalized and machine-checked in Lean 4 against Mathlib.