Dimension reduction for maximum matchings and the Fastest Mixing Markov Chain
Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta $ and vertex conductance $\Psi ^*(G)$. We show that there exists a symmetric, stochastic matrix $P$, with off-diagonal entries supported on $E$, whose spectral gap $\gamma ^*(P)$ satisfies \[ \Psi ^*(G)^{2}/\log \Delta \lesssim \gamm...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-07-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.447/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta $ and vertex conductance $\Psi ^*(G)$. We show that there exists a symmetric, stochastic matrix $P$, with off-diagonal entries supported on $E$, whose spectral gap $\gamma ^*(P)$ satisfies
\[ \Psi ^*(G)^{2}/\log \Delta \lesssim \gamma ^*(P) \lesssim \Psi ^*(G). \]
Our bound is optimal under the Small Set Expansion Hypothesis, and answers a question of Olesker-Taylor and Zanetti, who obtained such a result with $\log \Delta $ replaced by $\log |V|$.In order to obtain our result, we show how to embed a negative-type semi-metric $d$ defined on $V$ into a negative-type semi-metric $d^{\prime }$ supported in $\mathbb{R}^{O(\log \Delta )}$, such that the (fractional) matching number of the weighted graph $(V,E,d)$ is approximately equal to that of $(V,E,d^{\prime })$. |
|---|---|
| ISSN: | 1778-3569 |