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

Abstract

Buchbinder and Feldman recently gave a deterministic $(1-1/e-\varepsilon)$-approximation for maximizing a non-negative monotone submodular function subject to a matroid constraint, with query complexity $\widetilde{O}_\varepsilon(nr)$. Their algorithm uses an integer parameter $\ell$, which Buchbinder and Feldman fix to $\ell = 1 + \lceil 1/\varepsilon \rceil$ via a loose bound on $(1+1/\ell)^{-\ell}$. We point out two purely elementary refinements. First, the classical P\'olya--Szeg\H{o} inequality $(1+1/\ell)^{-\ell} \le e^{-1}(1+1/(2\ell))$ replaces the loose step in their proof and permits $\ell = \lceil 1/(2e\varepsilon) \rceil$, shrinking the hidden constant in $\widetilde{O}_\varepsilon(nr)$ by a factor $\approx 2^{0.816/\varepsilon}$. Second, an alternating-series tail bound for $\log(1+t)$ yields the asymptotically sharp inequality $(1+1/\ell)^{-\ell} \le e^{-1}\exp(1/(2\ell) - 1/(3\ell^2) + 1/(4\ell^3))$, matching the true expansion of $(1+1/\ell)^{-\ell}$ through order $\ell^{-3}$ and translating into $\ell_\star = 1/(2e\varepsilon) - 5/12 + O(\varepsilon)$. The asymptotic class $\widetilde{O}_\varepsilon(nr)$ of the query complexity is unchanged in either case; only the implicit constant in $\varepsilon$ is improved. All inequalities in this note are formalized and machine-checked in Lean 4 against Mathlib.