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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.08984 |
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Table of Contents:
- This paper studies the problem of inferring a $k$-factor, specifically a spanning $k$-regular graph, planted within an Erdos--Renyi random graph $G(n,λ/n)$. We uncover an interesting "all-something-nothing" phase transition. Specifically, we show that as the average degree $λ$ surpasses the critical threshold of $1/k$, the inference problem undergoes a transition from almost exact recovery ("all" phase) to partial recovery ("something" phase). Moreover, as $λ$ tends to infinity, the accuracy of recovery diminishes to zero, leading to the onset of the "nothing" phase. This finding complements the recent result by Mossel, Niles-Weed, Sohn, Sun, and Zadik who established that for certain sufficiently dense graphs, the problem undergoes an "all-or-nothing" phase transition, jumping from near-perfect to near-zero recovery. In addition, we characterize the recovery accuracy of a linear-time iterative pruning algorithm and show that it achieves almost exact recovery when $λ< 1/k$. A key component of our analysis is a two-step cycle construction: we first build trees through local neighborhood exploration and then connect them by sprinkling using reserved edges. Interestingly, for proving impossibility of almost exact recovery, we construct $Θ(n)$ many small trees of size $Θ(1)$, whereas for establishing the algorithmic lower bound, a single large tree of size $Θ(\sqrt{n\log n})$ suffices.