Motivated by applications in production planning and storage allocation in hierarchical databases, we initiate the study of covering partially ordered items (CPO). Given a capacity $k \in \mathbb{Z}^+$, and a directed graph $G=(V,E)$ where each vertex has a size in $\{0,1, \ldots,k\}$, we seek a collection of subsets of vertices $S_1, \ldots, S_m$ that cover all the vertices, such that for any $1 \leq j \leq m$, the total size of vertices in $S_j$ is bounded by $k$, and there are no edges from $V \setminus S_j$ to $S_j$. The objective is to minimize the number of subsets $m$. CPO is closely related to the rule caching problem (RCP) that is of wide interest in the networking area. The input for RCP is a directed graph $G=(V,E)$, a profit function $p:V \rightarrow \mathbb{Z}_{0}^+$, and $k \in \mathbb{Z}^+$. The output is a subset $S \subseteq V$ of maximum profit such that $|S| \leq k$ and there are no edges from $V \setminus S$ to $S$. Our main result is a $2$-approximation algorithm for CPO on out-trees, complemented by an asymptotic $1.5$-hardness of approximation result. We also give a two-way reduction between RCP and the densest $k$-subhypergraph problem, surprisingly showing that the problems are equivalent w.r.t. polynomial-time approximation within any factor $\rho \geq 1$. This implies that RCP cannot be approximated within factor $|V|^{1-\eps}$ for any fixed $\eps>0$, under standard complexity assumptions. Prior to this work, RCP was just known to be strongly NP-hard. We further show that there is no EPTAS for the special case of RCP where the profits are uniform, assuming Gap-ETH. Since this variant admits a PTAS, we essentially resolve the complexity status of this problem.

翻译：受生产规划与层次化数据库存储分配应用的启发，我们首次研究了部分有序物品覆盖问题（CPO）。给定正整数容量 $k \in \mathbb{Z}^+$ 及有向图 $G=(V,E)$（其中每个顶点的大小属于 $\{0,1,\ldots,k\}$），需寻找覆盖所有顶点的子集族 $S_1,\ldots,S_m$，满足：对任意 $1 \leq j \leq m$，$S_j$ 中顶点总大小不超过 $k$，且不存在从 $V \setminus S_j$ 到 $S_j$ 的边。优化目标为最小化子集数量 $m$。CPO 与网络领域广泛关注的规则缓存问题（RCP）密切相关。RCP 的输入为有向图 $G=(V,E)$、利润函数 $p:V \rightarrow \mathbb{Z}_{0}^+$ 及 $k \in \mathbb{Z}^+$，输出为满足 $|S| \leq k$ 且无 $V\setminus S$ 到 $S$ 边的最大利润子集 $S \subseteq V$。本文主要结果为：针对外向树结构给出了 CPO 的 2-近似算法，并证明了渐近 1.5 的近似难度下界。我们同时建立了 RCP 与最密 $k$-子超图问题的双向归约，意外表明两者在任意近似因子 $\rho \geq 1$ 下关于多项式时间近似是等价的。这一结论暗示：在标准复杂性假设下，RCP 不存在 $|V|^{1-\eps}$ 因子近似（对任意固定 $\eps>0$）。此前 RCP 仅被证明为强 NP-难问题。我们进一步证明：在 Gap-ETH 假设下，利润均匀的 RCP 特例不存在有效多项式时间近似方案（EPTAS）。鉴于该变体存在多项式时间近似方案（PTAS），本研究实质上解决了该问题的复杂性状态。