发表机构
Syracuse University（雪城大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对未知图从顶点删除痕迹重建度序列的问题，提出两种算法，分别基于拒绝采样和度矩恢复，并给出边数恢复的复杂度上下界。
AI 中文摘要
度序列重建的目标是从顶点删除痕迹中恢复未知图的有序度向量，其中每个顶点以概率$p$独立删除。我们提供了两种重建算法；第一种使用拒绝采样将问题简化为混合分布的估计问题。结合先前的痕迹重建结果，这给出了一种使用$\tilde O(n^{1/3})$条痕迹的重建算法，尽管没有已知的次指数时间解码器。我们的另一种方法涉及恢复某些图不变量，即度矩，这些矩可以识别图的度序列。极值多项式界表明，$\tilde \theta(n^{1/2})$个度矩对于重建度序列是必要且充分的，这导致了一种具有$\tilde O(n^{1/2})$痕迹复杂度的算法。相同的多项式机制产生了一种从矩中恢复度序列的次指数时间解码器。此外，我们给出了恢复边数的痕迹复杂度的$O(n^{3})$上界和$\theta(n^2)$下界。
英文摘要
The goal of degree sequence reconstruction is to recover the ordered vector of degrees of an unknown graph from vertex deleted traces, where each vertex is deleted independently with probability $p$. We provide two algorithms for reconstruction; the first uses rejection sampling to reduce the problem to an estimation problem for a mixture distribution. Combined with prior trace reconstruction results, this gives a reconstruction algorithm using $\Exp{\tilde O (n^{1/3})}$ traces, although no sub-exponential time decoder is known. Our other approach involves recovering certain graph invariants, degree moments, that can identify a graphs degree sequence. Extremal polynomial bounds show that $\tilde Θ(n^{1/2})$ degree moments are necessary and sufficient to reconstruct the degree sequence, which leads to an algorithm with $\Exp{\tilde O(n^{1/2})}$ trace complexity. The same polynomial machinery yields a sub-exponential time decoder for the degree sequence from the moments. Additionally, we give an $O(n^{3})$ upper bound and a $Ω(n^2)$ lower bound for the trace complexity of recovering the number of edges.
Comments35 pages, submitted to ITCS 2026
