Close the Edge-Count Reconstruction Sample-Complexity Gap

Close the gap between the $O(n^3)$ upper bound and the $Omega(n^2)$ lower bound for the number of vertex-deletion traces required to reconstruct the number of edges of an $n$-vertex graph.

Background

For recovering the number of edges, the paper proves an $O(n^3)$ upper bound and an $Omega(n^2)$ lower bound on the required number of degree-sequence traces. The bounds differ by a factor of order $n$.

The authors explicitly classify closing this gap as an open problem related to graph trace reconstruction.