On Bounds for Greedy Schemes in String Optimization based on Greedy Curvatures
Bowen Li, Brandon Van Over, Edwin K. P. Chong, Ali Pezeshki
TL;DR
This work extends the classical greedy-approximation analysis from set submodularity to string optimization, where order matters and the horizon matters. It derives a computable bound $\beta_{1}$ based on a generalized greedy curvature under assumptions $\mathbf{A_1}$–$\mathbf{A_3}$, and a stronger bound $\beta_{2}$ under $\mathbf{A_1}$–$\mathbf{A_2}$, with $\beta_{2} \ge \beta_{1}$. Crucially, these bounds are computable along the greedy trajectory and require weaker conditions than full string submodularity, providing tighter performance guarantees than prior curvature-based bounds in many cases. The authors illustrate the practical impact with task scheduling and multi-agent sensor coverage, showing that the new bounds can outperform the classic $(1-e^{-1})$ benchmark and are readily evaluable in real-time settings.
Abstract
We consider the celebrated bound introduced by Conforti and Cornuéjols (1984) for greedy schemes in submodular optimization. The bound assumes a submodular function defined on a collection of sets forming a matroid and is based on greedy curvature. We show that the bound holds for a very general class of string problems that includes maximizing submodular functions over set matroids as a special case. We also derive a bound that is computable in the sense that they depend only on quantities along the greedy trajectory. We prove that our bound is superior to the greedy curvature bound of Conforti and Cornuéjols. In addition, our bound holds under a condition that is weaker than submodularity.