Asymptotically Optimal Hardness for $k$-Set Packing and $k$-Matroid Intersection
Euiwoong Lee, Ola Svensson, Theophile Thiery
TL;DR
The paper proves an asymptotically tight hardness of approximation for $k$-Dimensional Matching and related problems, showing that unless ${\sf NP} \subseteq {\sf BPP}$, no polynomial-time algorithm can achieve better than a factor of $k/(12+\varepsilon)$ for large constant $k$. The approach centers on an approximation-preserving gadget that maps an $R$-degree bounded $k$-CSP over alphabet $R$ to a $kR$-Dimensional Matching, and a bounded-degree sparsification technique that preserves completeness while tightening the soundness to $s' = \frac{k(1+\lambda)}{(\gamma-1)(1-\lambda)^2 R}$. By iterating these steps, the authors obtain a hardness transfer from bounded-degree CSPs to $kR$-DM and then lift it to $p$-DM for $p=kR$, yielding a constant-factor inapproximability independent of $k$ for large $k$. Consequently, $k$-Set Packing, $k$-Matroid Intersection, $k$-Matchoid, and $k$-Matroid Parity inherit the same $k/12$-hardness up to the $\varepsilon$ term, clarifying why algorithmic progress beyond linear in $k$ remains elusive. The results provide a principled explanation for observed algorithmic difficulties and establish a framework for extending hardness via degree-bounded CSP sparsification.
Abstract
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 $Ω(k /\log(k))$-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 $Ω_k(R)$ using known randomised sparsification methods for CSPs.