一般组合约束下非自适应随机Top-k和问题的PTAS
A PTAS for Non-Adaptive Stochastic Top-$k$ Sum under General Combinatorial Constraints
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中文总结 AI 辅助
该研究针对一般组合约束下的非自适应随机Top-k和问题，利用近似最大和预言机与打包LP等技术，为固定维打包族（含背包问题）构造了PTAS，并证明其为通用最优方案
中文摘要 AI 辅助
我们研究非自适应选择一个可行集S，以最大化独立非负离散随机变量中k个最大实现值的期望和。当雇佣k名工人或在l单位拍卖中计算VCG福利时，会出现相同的目标。主要设定是固定维非负打包族：自然线性规划（LP）具有d=O(1)个带二进制系数的打包不等式。不存在单一算法能在每个成员族上都达到常数因子（即使k=1时也是如此）。给定一个α近似最大和预言机，递减盈余搜索对每个k≥1（包括割）都能得到比率α/((1+α)(1+ε))。每个固定d的打包族已存在确定性最大和PTAS，因此可继承该常数。驱动精确和方案的相同特征——当k=O(1/ε²)时的占用直方图，以及当k=Ω(1/ε²)时的三维混合分位数类型——是通过枚举n^{f(d,1/ε)}个重项，由打包LP而非精确和实现的。结果是，在每个固定d的打包族（包括二进制一维和二维背包）上，对每个k≥1都存在PTAS。在该打包设定下，该方案作为通用保证本质上是最优的：除非P=NP，否则不存在适用于每个此类族F的FPTAS；除非W[1]=FPT，否则不存在EPTAS（二维背包是佐证，即使k=1时也是如此）。另一个边界是查询权重精确和，包括有向无环图（DAG）路径和匹配，且与固定d打包不可比。该预言机也能为每个k生成PTAS，因此d维打包是有用的分类法，而非每个允许PTAS的族的划分。
英文摘要
We study non-adaptive selection of a feasible set $S$ that maximizes the expected sum of the $k$ largest realized values among independent nonnegative discrete random variables. The same objective arises in team hiring and as VCG welfare in an $\ell$-unit auction. The main setting is a fixed-dimensional nonnegative packing family, whose natural LP has $d=O(1)$ packing inequalities with binary coefficients. We give a PTAS for every $k\ge 1$ on every such family, including binary one- and two-dimensional knapsack, by approximating the occupancy functional $p\mapsto\mathbb{E}[\min(k,N(p))]$ and realizing the resulting signatures in the packing LP. As a generic guarantee the scheme is essentially optimal: there is no FPTAS that works for every such $\mathcal{F}$ unless $P=NP$, and no EPTAS unless $W[1]=FPT$. An incomparable sufficient condition is a query-weight exact-sum oracle (DAG paths, matchings), which likewise yields a PTAS for every $k$. The same signatures give a PTAS for $\min_{S\in\mathcal{F}}\mathbb{E}[\mathrm{Top}_k(S)]$ on every fixed-$d$ covering family; two-dimensional covering knapsack rules out a generic FPTAS on that class.
