发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于支持区域几何的线性化方法，为非单调 $k$-子模最大化提供改进的认证近似系数，并推广到拟阵、背包及在线场景。
AI 中文摘要
我们研究了在支持约束下具有 $k\ge2$ 个标签的非负、非单调 $k$-子模最大化问题，并展示了当支持区域允许更均匀的选择时，认证近似系数如何改善。对于紧致凸向下封闭的支持区域 $P\subseteq[0,1]^n$，对角线水平 $\zeta(P)=\max\{t\in[0,1]:t {\bf 1} \in P\}$ 的取值范围从 $\zeta=0$（不携带任何几何保证）到 $\zeta=1$（无限制支持）。我们的主要结构结果是多线性扩展的 comparator-uniform 线性化，由独立于目标的作用和独立于 comparator 的更新场构建。对于 $k\ge3$，其有效性独立于元素数量，简化为四个次数至多为三的一元或二元多项式不等式，其中仅一个依赖于 $k$。显式参数选择给出了非递减的认证轮廓 $\underline{\alpha}_k(\zeta)$，当 $k=2$ 时在 $[0,1]$ 上具有闭式形式。在 $\zeta=0$ 处，我们对 $k=2$ 认证 $0.4456\ldots$，对每个 $k\ge3$ 认证 $0.4541\ldots$，改进了最近针对一个拟阵或一个背包的 $\sqrt2-1$ 保证，以及针对固定数量预算的 $1/3$ 型保证；在 $\zeta=1$ 处，我们对 $k=2$ 认证 $1/2$，对 $k=3,4$ 认证 $(\sqrt{17}-3)/2$，对 $k\ge5$ 认证 $k/(2k-1)$，其超过 $1/2$ 的部分为 $1/k$ 阶而非之前的 $1/k^2$ 阶。保持值的舍入将这些保证转移到拟阵和背包约束，相同的场在梯度或决策后值反馈下在线产生 $O(\sqrt T)$ 的近似遗憾。
英文摘要
We study nonnegative, non-monotone $k$-submodular maximization with $k\ge2$ labels under support constraints, and show how the certified approximation coefficient improves as the support region permits more uniform selection. For a compact convex down-closed support region $P\subseteq[0,1]^n$, the diagonal level $ζ(P)=\max\{t\in[0,1]:t {\bf 1} \in P\}$ ranges from $ζ=0$, which carries no geometric promise, to $ζ=1$, which is unrestricted support. Our main structural result is a comparator-uniform linearization of the multilinear extension, built from an objective-independent action and a comparator-independent update field. For $k\ge3$, its validity reduces, independently of the number of elements, to four polynomial inequalities of degree at most three in one or two variables, only one of which depends on $k$. Explicit parameter choices give a nondecreasing certified profile $\underlineα_k(ζ)$, in closed form on all of $[0,1]$ when $k=2$. At $ζ=0$ we certify $0.4456\ldots$ for $k=2$ and $0.4541\ldots$ for every $k\ge3$, improving the recent $\sqrt2-1$ guarantee for one matroid or one knapsack, as well as the $1/3$-type guarantees for a fixed number of budgets; at $ζ=1$ we certify $1/2$ for $k=2$, $(\sqrt{17}-3)/2$ for $k=3,4$, and $k/(2k-1)$ for $k\ge5$, whose excess over $1/2$ is of order $1/k$ rather than the previous $1/k^2$. Value-retaining rounding transfers these guarantees to matroid and knapsack constraints, and the same field yields $O(\sqrt T)$ approximate regret online under gradient or post-decision value feedback.
