Lower bound for edge counting under local node differential privacy

Establish the Omega_k(n^{k-1/2}) lower bound for noninteractive local node-differentially-private algorithms that count copies of a template graph with maximum degree equal to 1, thereby resolving the corresponding open edge-counting lower bound, which remains equivalent even for templates with more than two nodes.

Background

The paper proves an Omega_k(n^{k-1/2}) lower bound for counting copies of any constant-size template graph whose maximum degree is greater than 1 under noninteractive local node differential privacy. The authors explicitly note that extending the same result to templates with maximum degree equal to 1 would imply an open lower bound for edge counting identified in prior work.

A template with maximum degree equal to 1 is a disjoint union of isolated vertices and edges. The note further observes that, even when such a template has more than two nodes, its counting problem is equivalent to edge counting for purposes of this lower bound. Thus, the unresolved task is to obtain the analogous optimal lower bound for edge counting under local node differential privacy.