Rapid mixing of the down-up walk on matchings of a fixed size
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929451290329088 |
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| author | Jain, Vishesh Mizgerd, Clayton |
| author_facet | Jain, Vishesh Mizgerd, Clayton |
| contents | Let $G = (V,E)$ be a graph on $n$ vertices and let $m^*(G)$ denote the size of a maximum matching in $G$. We show that for any $δ> 0$ and for any $1 \leq k \leq (1-δ)m^*(G)$, the down-up walk on matchings of size $k$ in $G$ mixes in time polynomial in $n$. Previously, polynomial mixing was not known even for graphs with maximum degree $Δ$, and our result makes progress on a conjecture of Jain, Perkins, Sah, and Sawhney [STOC, 2022] that the down-up walk mixes in optimal time $O_{Δ,δ}(n\log{n})$.
In contrast with recent works analyzing mixing of down-up walks in various settings using the spectral independence framework, we bound the spectral gap by constructing and analyzing a suitable multi-commodity flow. In fact, we present constructions demonstrating the limitations of the spectral independence approach in our setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03466 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rapid mixing of the down-up walk on matchings of a fixed size Jain, Vishesh Mizgerd, Clayton Data Structures and Algorithms Combinatorics Let $G = (V,E)$ be a graph on $n$ vertices and let $m^*(G)$ denote the size of a maximum matching in $G$. We show that for any $δ> 0$ and for any $1 \leq k \leq (1-δ)m^*(G)$, the down-up walk on matchings of size $k$ in $G$ mixes in time polynomial in $n$. Previously, polynomial mixing was not known even for graphs with maximum degree $Δ$, and our result makes progress on a conjecture of Jain, Perkins, Sah, and Sawhney [STOC, 2022] that the down-up walk mixes in optimal time $O_{Δ,δ}(n\log{n})$. In contrast with recent works analyzing mixing of down-up walks in various settings using the spectral independence framework, we bound the spectral gap by constructing and analyzing a suitable multi-commodity flow. In fact, we present constructions demonstrating the limitations of the spectral independence approach in our setting. |
| title | Rapid mixing of the down-up walk on matchings of a fixed size |
| topic | Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2408.03466 |