Asymptotically Optimal Hardness for $k$-Set Packing and $k$-Matroid Intersection
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arXiv
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| Natura: | Preprint |
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2024
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| author | Lee, Euiwoong Svensson, Ola Thiery, Theophile |
| author_facet | Lee, Euiwoong Svensson, Ola Thiery, Theophile |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17831 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotically Optimal Hardness for $k$-Set Packing and $k$-Matroid Intersection Lee, Euiwoong Svensson, Ola Thiery, Theophile Computational Complexity Data Structures and Algorithms Combinatorics 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. |
| title | Asymptotically Optimal Hardness for $k$-Set Packing and $k$-Matroid Intersection |
| topic | Computational Complexity Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2409.17831 |