发表机构
Nanyang Technological University（南洋理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明一致子模最大化中，多项式查询与常数补偿的近似比上确界为 $2-\sqrt2$，任何改进需指数查询或线性补偿，并给出匹配算法及曲率相关阈值。
AI 中文摘要
一致子模最大化研究当元素随时间到达时，解质量与稳定性之间的权衡。对于单调子模目标函数（其建模了收益递减），算法维护一个最多包含 $k$ 个可用元素的集合，并在每次插入后仅改变 $O(1)$ 个元素。Dütting 等人 [2025] 在无限制计算下建立了紧的 $2/3$ 近似比，并给出了多项式时间的 $0.51$ 近似算法。他们在 STOC 2025 上留下了开放问题：高效算法能否匹配离线 $1-1/e$ 的保证。我们通过证明使用多项式次值查询和最坏情况常数补偿可达到的近似比上确界为 \\[ \beta=2-\sqrt2\approx0.5858<1-1/e \\] 来解决该问题。对于每个 $\varepsilon>0$，我们的随机化算法以每次插入 $O(\varepsilon^{-2})$ 次改变达到 $\beta-\varepsilon$。任何固定的改进都需要在某个关键插入之前进行指数级查询，或在该插入时具有 $\Omega(k)$ 次改变的线性补偿，即使之后允许无限查询。这一差距量化了一致性的代价：当前的预言机隐藏了在到达后哪些元素将被需要。我们还确定了精确的曲率相关阈值 $1-(\sqrt2-1)\vartheta$，对于加权覆盖问题以 $O(\varepsilon^{-1})$ 补偿达到 $1-1/e-\varepsilon$，并区分了通用未来价格证书的存在性与其高效计算。我们的算法对于多项式位有理数预言机答案具有有界位多项式时间实现；下界仅使用对数位有理数答案。
英文摘要
Consistent submodular maximization studies the tradeoff between solution quality and stability when elements arrive over time. For a monotone submodular objective, which models diminishing returns, an algorithm maintains a set of at most $k$ available elements and changes only $O(1)$ elements after each insertion. Dütting et al. [2025] established a tight $2/3$ approximation with unrestricted computation and a polynomial-time $0.51$ approximation. They left open at STOC 2025 whether efficient algorithms can match the offline $1-1/e$ guarantee. We resolve this problem by proving that the supremum approximation achievable with polynomially many value queries and worst-case constant recourse is \[ β=2-\sqrt2\approx0.5858<1-1/e. \] For every $\varepsilon>0$, our randomized algorithm attains $β-\varepsilon$ with $O(\varepsilon^{-2})$ changes per insertion. Any fixed improvement requires exponentially many queries before one critical insertion or linear recourse of $Ω(k)$ changes at that insertion, even with unlimited queries afterwards. This gap quantifies the cost of consistency: the current oracle hides which elements will be needed after an arrival. We also determine the exact curvature-dependent threshold $1-(\sqrt2-1)\vartheta$, attain $1-1/e-\varepsilon$ for weighted coverage with $O(\varepsilon^{-1})$ recourse, and separate the existence of universal future-price certificates from their efficient computation. Our algorithm has a bounded-bit polynomial-time implementation for polynomial-bit rational oracle answers; the lower bound uses only logarithmic-bit rational answers.
