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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.11359 |
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| _version_ | 1866909863004602368 |
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| author | Han, Jie Zhao, Jingwen |
| author_facet | Han, Jie Zhao, Jingwen |
| contents | The decision problem of perfect matchings in uniform hypergraphs is famously an NP-complete problem. It has been shown by Keevash--Knox--Mycroft [STOC, 2013] that for every $\varepsilon>0$, such decision problem restricted to $k$-uniform hypergraphs $H$ satisfying that every $(k-1)$-set of vertices is in at least $(1/k+\varepsilon)|H|$ edges is tractable, and the quantity $1/k$ is best possible. In this paper we study the existence of perfect matchings in the random $p$-sparsification of such $k$-uniform hypergraphs, that is, for $p=p(n)\in [0,1]$, every edge is kept with probability $p$ independent of others. As a consequence, we give a polynomial-time algorithm that with high probability solves the decision problem; we also derive effective bounds on the number of perfect matchings in such hypergraphs. At last, similar results are obtained for the $F$-factor problem in graphs.
The key ingredients of the proofs are a strengthened partition lemma for the lattice-based absorption method, and the random redistribution method developed recently by Kelly, Müyesser and Pokrovskiy, based on the spread method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11359 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perfect Matchings in Random Sparsifications of Dense Hypergraphs Han, Jie Zhao, Jingwen Combinatorics The decision problem of perfect matchings in uniform hypergraphs is famously an NP-complete problem. It has been shown by Keevash--Knox--Mycroft [STOC, 2013] that for every $\varepsilon>0$, such decision problem restricted to $k$-uniform hypergraphs $H$ satisfying that every $(k-1)$-set of vertices is in at least $(1/k+\varepsilon)|H|$ edges is tractable, and the quantity $1/k$ is best possible. In this paper we study the existence of perfect matchings in the random $p$-sparsification of such $k$-uniform hypergraphs, that is, for $p=p(n)\in [0,1]$, every edge is kept with probability $p$ independent of others. As a consequence, we give a polynomial-time algorithm that with high probability solves the decision problem; we also derive effective bounds on the number of perfect matchings in such hypergraphs. At last, similar results are obtained for the $F$-factor problem in graphs. The key ingredients of the proofs are a strengthened partition lemma for the lattice-based absorption method, and the random redistribution method developed recently by Kelly, Müyesser and Pokrovskiy, based on the spread method. |
| title | Perfect Matchings in Random Sparsifications of Dense Hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.11359 |