Submodular Ground-Set Pruning: Monotone Tightness and a Non-Monotone Separation

TL;DR

This paper investigates containment pruning for large-scale submodular maximization, establishing tight bounds for monotone cases and novel algorithms for non-monotone objectives, with empirical benefits demonstrated on MaxCut and LLM tasks.

Contribution

It provides the first theoretical bounds for containment pruning, proves that non-monotone cases are easier than optimization, and introduces practical algorithms with empirical validation.

Findings

Greedy achieves the tight $1-1/e$ containment factor for monotone submodular functions.

Non-monotone objectives admit $1/2- ext{epsilon}$ containment algorithms, surpassing previous ratios.

Pruning significantly accelerates MaxCut solving and benefits LLM context selection.

Abstract

Large-scale subset selection asks for a small useful set of examples, features, sensors, seed users, or context passages from an enormous ground set. Submodular maximization is a canonical model for such diminishing-returns problems, but rapidly growing datasets make even linear-time algorithms ever costlier. We study \emph{containment pruning}: first reduce the ground set to a smaller core $P$, then require that $P$ contain a near-optimal feasible solution for every downstream budget up to~$k$. Prior work has formulated many heuristics, but the theoretical limits of this preprocessing problem are largely unknown. For monotone submodular objectives, we prove that $1-1/e$ is tight: greedy achieves this containment factor, and no algorithm can beat it even with a larger pruning budget. For non-monotone objectives, we give the first$1/2-\varepsilon$ containment algorithms under cardinality constraints and extend the approach to knapsack constraints. This $1/2$ factor exceeds the best known algorithmic ratio and the known hardness threshold for non-monotone maximization, showing that pruning can be provably easier than optimization. Empirically, pruning lets an exact IP solver run on the reduced MaxCut instance with a ${\approx}620\times$ speedup, and proof-of-concept experiments on LLM context selection demonstrate the utility of non-monotone submodular proxies and our proposed containment algorithms.