We study the $d$-dimensional knapsack problem. We are given a set of items, each with a $d$-dimensional cost vector and a profit, along with a $d$-dimensional budget vector. The goal is to select a set of items that do not exceed the budget in all dimensions and maximize the total profit. A PTAS with running time $n^{\Theta(d/\varepsilon)}$ has long been known for this problem, where $\varepsilon$ is the error parameter and $n$ is the encoding size. Despite decades of active research, the best running time of a PTAS has remained $O(n^{\lceil d/\varepsilon \rceil - d})$. Unfortunately, existing lower bounds only cover the special case with two dimensions $d = 2$, and do not answer whether there is a $n^{o(d/\varepsilon)}$-time PTAS for larger values of $d$. The status of exact algorithms is similar: there is a simple $O(n \cdot W^d)$-time (exact) dynamic programming algorithm, where $W$ is the maximum budget, but there is no lower bound which explains the strong exponential dependence on $d$. In this work, we show that the running times of the best-known PTAS and exact algorithm cannot be improved up to a polylogarithmic factor assuming Gap-ETH. Our techniques are based on a robust reduction from 2-CSP, which embeds 2-CSP constraints into a desired number of dimensions, exhibiting tight trade-off between $d$ and $\varepsilon$ for most regimes of the parameters. Informally, we obtain the following main results for $d$-dimensional knapsack. No $n^{o(d/\varepsilon \cdot 1/(\log(d/\varepsilon))^2)}$-time $(1-\varepsilon)$-approximation for every $\varepsilon = O(1/\log d)$. No $(n+W)^{o(d/\log d)}$-time exact algorithm (assuming ETH). No $n^{o(\sqrt{d})}$-time $(1-\varepsilon)$-approximation for constant $\varepsilon$. $(d \cdot \log W)^{O(d^2)} + n^{O(1)}$-time $\Omega(1/\sqrt{d})$-approximation and a matching $n^{O(1)}$-time lower~bound.

翻译：本文研究$d$维背包问题。给定一组物品，每个物品具有$d$维代价向量和利润值，同时给定$d$维预算向量。目标是在所有维度上均不超出预算的前提下，选择一组物品以最大化总利润。长期以来，针对该问题已知存在运行时间为$n^{\Theta(d/\varepsilon)}$的PTAS，其中$\varepsilon$为误差参数，$n$为编码规模。尽管经过数十年的深入研究，PTAS的最佳运行时间始终停留在$O(n^{\lceil d/\varepsilon \rceil - d})$。遗憾的是，现有下界仅涵盖二维特殊情形（$d=2$），未能回答对于更大的$d$值是否存在$n^{o(d/\varepsilon)}$时间的PTAS。精确算法的研究现状也类似：存在简单的$O(n \cdot W^d)$时间（精确）动态规划算法，其中$W$为最大预算值，但尚无下界能够解释对$d$的强指数依赖关系。本工作证明，在Gap-ETH假设下，已知最优PTAS与精确算法的运行时间无法改进至多对数因子。我们的技术基于从2-CSP出发的鲁棒归约，通过将2-CSP约束嵌入指定维度数，在多数参数范围内展现出$d$与$\varepsilon$之间的严格权衡关系。非正式地，我们为$d$维背包问题获得以下主要结果：对于任意$\varepsilon = O(1/\log d)$，不存在$n^{o(d/\varepsilon \cdot 1/(\log(d/\varepsilon))^2)}$时间的$(1-\varepsilon)$近似算法；在ETH假设下，不存在$(n+W)^{o(d/\log d)}$时间的精确算法；对于常数$\varepsilon$，不存在$n^{o(\sqrt{d})}$时间的$(1-\varepsilon)$近似算法；存在$(d \cdot \log W)^{O(d^2)} + n^{O(1)}$时间的$\Omega(1/\sqrt{d})$近似算法，并给出匹配的$n^{O(1)}$时间下界。