发表机构
Faculty of Management, University of New Brunswick（新不伦瑞克大学管理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对DR-次模多项式最大化问题，提出基于凹屋顶与向下恢复的确定性多项式时间近似算法，分别取得二次情形1/2、三次情形8/17的近似因子，并推广至更广函数类。
AI 中文摘要
我们针对问题 $\max\{F(x):x\in\K\}$ 给出近似算法，其中 $\K$ 是单位立方体中的非空紧致向下闭凸子集，$F$ 是非负递减回报（DR）次模多项式：其 Hessian 矩阵在立方体上的每个元素均为非正。从二次或三次系数出发，我们将目标函数精确分解为非负根原子项与仿射项之和。这些分量的凹上界给出了一个易处理的屋顶（roof）。随后，我们使用对所有分量同时有效的映射来降低屋顶优化器的坐标，在保持可行性的同时恢复屋顶值的一部分。对于二次情形，我们还精确保留凹对角残差。在所述优化访问假设下，这些构造给出了确定性的多项式时间因子：二次情形为 $1/2$，三次情形为 $8/17$，误差可达任意指定的加法误差。三次结果包含正三次系数和重复变量。因子 $8/17$ 也适用于两个更大的类别：提供的兼容根证书（字面次数至多为十二）以及至多四元的非负次模局部函数之和。在任意有限字面次数下，一个公共标量映射达到最优单映射因子 $2\sqrt3-3$。全局单调性和凹基数结构为多线性局部函数和提供了更强的保证，且无需超出恒等映射的恢复。标准拟阵舍入可无损失地转移多线性保证。我们还证明了有向超图割的在线保持性质，并分别证明了在基数上界下，显式二次输入在唯一博弈（Unique Games）下的不可近似性界限约为 $0.82843$。
英文摘要
We give approximation algorithms for $\max\{F(x):x\in\K\}$, where $\K$ is a nonempty compact down-closed convex subset of the unit cube and $F$ is a nonnegative diminishing-returns (DR) submodular polynomial: every entry of its Hessian is nonpositive on the cube. Starting from quadratic or cubic coefficients, we decompose the objective exactly into nonnegative rooted atoms and affine terms. Concave upper bounds for these components give a tractable roof. We then decrease the coordinates of a roof optimizer using maps that work for all components at once, recovering a fraction of the roof value while preserving feasibility. For quadratics, we also retain the concave diagonal residual exactly. Under the stated assumptions on optimization access, these constructions give deterministic polynomial-time factors of $1/2$ for quadratics and $8/17$ for cubics, up to any prescribed additive error. The cubic result includes positive cubic coefficients and repeated variables. The factor $8/17$ also holds for two larger classes: supplied compatible rooted certificates of literal degree at most twelve, and sums of nonnegative submodular local functions of arity at most four. At arbitrary finite literal degree, a common scalar map attains the optimal single-map factor $2\sqrt3-3$. Global monotonicity and concave-cardinality structure give stronger guarantees for multilinear local-function sums, with no recovery beyond the identity map. Standard matroid rounding transfers the multilinear guarantees without loss. We also prove online preservation for directed-hypergraph cuts and, separately, a Unique-Games inapproximability bound of approximately $0.82843$ for explicit quadratic input under a cardinality upper bound.
