For any $\varepsilon > 0$, we prove that $k$-Dimensional Matching is hard to approximate within a factor of $k/(12 + \varepsilon)$ for large $k$ unless $\textsf{NP} \subseteq \textsf{BPP}$. Listed in Karp's 21 $\textsf{NP}$-complete problems, $k$-Dimensional Matching is a benchmark computational complexity problem which we find as a special case of many constrained optimization problems over independence systems including: $k$-Set Packing, $k$-Matroid Intersection, and Matroid $k$-Parity. For all the aforementioned problems, the best known lower bound was a $\Omega(k /\log(k))$-hardness by Hazan, Safra, and Schwartz. In contrast, state-of-the-art algorithms achieved an approximation of $O(k)$. Our result narrows down this gap to a constant and thus provides a rationale for the observed algorithmic difficulties. The crux of our result hinges on a novel approximation preserving gadget from $R$-degree bounded $k$-CSPs over alphabet size $R$ to $kR$-Dimensional Matching. Along the way, we prove that $R$-degree bounded $k$-CSPs over alphabet size $R$ are hard to approximate within a factor $\Omega_k(R)$ using known randomised sparsification methods for CSPs.

翻译：对于任意$\varepsilon > 0$，我们证明：除非$\textsf{NP} \subseteq \textsf{BPP}$，否则对于充分大的$k$，$k$维匹配问题在$k/(12 + \varepsilon)$的近似因子内是难以逼近的。作为Karp 21个$\textsf{NP}$完全问题之一的$k$维匹配问题，是一个基准计算复杂度问题。我们发现该问题可视为许多基于独立性系统的约束优化问题的特例，包括：$k$集合包装、$k$拟阵交以及拟阵$k$奇偶问题。对于所有上述问题，Hazan、Safra和Schwartz所证明的$\Omega(k /\log(k))$硬度下界是此前已知的最佳结果。相比之下，当前最先进的算法仅能实现$O(k)$的近似比。我们的研究将这一差距缩小至常数级别，从而为观察到的算法困难提供了理论依据。我们结果的核心在于构建了一种新颖的近似保持规约结构，该结构将字母表规模为$R$的$R$度有界$k$-CSP问题转化为$kR$维匹配问题。在此过程中，我们利用已知的CSP随机稀疏化方法，证明了字母表规模为$R$的$R$度有界$k$-CSP问题在$\Omega_k(R)$的近似因子内是难以逼近的。