arXiv:2609.09986 · v2

A Sharp Barrier of Consistent Submodular Maximization: Any Improvement over $2-\sqrt{2}$ Entails Exponential Queries or Linear Recourse

首次提交 2026-09-09 18:13

版本更新 2026-11-03 21:53

雷达观测 2026-11-04 23:39

摘要

Consistent submodular maximization aims to maintain a high-utility subset of size at most $k$ as new elements arrive one by one, while changing only a limited number of elements per update. Previously, [DFL+25](STOC '25) established an information-theoretic bound of $2/3$ and gave the first polynomial-time randomized algorithm with a $0.51$-approximation guarantee. However, a fundamental question remained open: Is it possible to design an efficient algorithm for consistent submodular maximization that matches the classical offline $(1-1/e)$-approximation? In this paper, we answer this question in the negative by proving that the sharp approximation threshold achievable with polynomially many value queries and worst-case constant recourse is \[β:=2-\sqrt{2}\approx0.5858<1-1/e.\] Specifically, we develop an improved poly-time randomized algorithm, which can attain a $(β-\varepsilon)$-approximation in expectation at each fixed time with $O(\varepsilon^{-2})$ worst-case recourse. The cornerstone of this algorithm is a novel anchored sampling scheme, which uses only value queries on subsets of the elements seen so far to construct a solution that remains competitive under unknown future arrivals. Conversely, we establish a matching lower bound by showing that any fixed improvement over $β$ requires either exponentially many value queries or $Ω(k)$ changes per update. Furthermore, we characterize the impact of curvature by establishing a sharp threshold of $1-(\sqrt{2}-1)\vartheta$ for consistent submodular maximization, where $\vartheta\in[0,1]$ denotes the total curvature. Surprisingly, for weighted coverage objectives, we devise a poly-time randomized algorithm that can match the optimal $(1-1/e-\varepsilon)$-approximation with $O(\varepsilon^{-1})$ worst-case recourse, thereby separating coverage maximization from general submodular maximization in the consistent settings.