Lower bounds for graph reconstruction with maximal independent set queries

Abstract

We investigate the number of maximal independent set queries required to reconstruct the edges of a hidden graph. We show that randomised adaptive algorithms need at least $\Omega ({\Delta }^2\log (n/\Delta )/\log \Delta )$ queries to reconstruct n-vertex graphs of maximum degree Δ with success probability at least 1/2, and we further improve this lower bound to $\Omega ({\Delta }^2\log (n/\Delta \left)\right)$ for randomised non-adaptive algorithms. We also prove that deterministic non-adaptive algorithms require at least $\Omega ({\Delta }^3\log n/\log \Delta )$ queries.

This improves bounds of Konrad, O'Sullivan, and Traistaru, and answers one of their questions. The proof of the lower bound for deterministic non-adaptive algorithms relies on a connection to cover-free families, for which we also improve known bounds.