Degree Sequence Reconstruction from Subgraph Traces

Abstract: 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 $\Exp{\tilde O (n<sup>{1/3})}$ 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 $\tilde Θ(n<sup>{1/2})$ degree moments are necessary and sufficient to reconstruct the degree sequence, which leads to an algorithm with $\Exp{\tilde O(n<sup>{1/2})}$ trace complexity. The same polynomial machinery yields a sub-exponential time decoder for the degree sequence from the moments. Additionally, we give an $O(n<sup>{3})$ upper bound and a $Ω(n<sup>2)$ lower bound for the trace complexity of recovering the number of edges.