Tight query complexity bounds for learning graph partitions
This work addresses query complexity in graph learning, providing tight bounds that refine theoretical understanding for researchers in algorithms and computational learning theory.
The paper proves that learning the connected components of a hidden graph with n vertices and k components requires at least (k-1)n - binom(k,2) membership queries, matching the complexity of an existing algorithm and improving on prior bounds. It also introduces new oracles for learning the number of components and for handling m-edge graphs, with tight query bounds.
Given a partition of a graph into connected components, the membership oracle asserts whether any two vertices of the graph lie in the same component or not. We prove that for $n\ge k\ge 2$, learning the components of an $n$-vertex hidden graph with $k$ components requires at least $(k-1)n-\binom k2$ membership queries. Our result improves on the best known information-theoretic bound of $Ω(n\log k)$ queries, and exactly matches the query complexity of the algorithm introduced by [Reyzin and Srivastava, 2007] for this problem. Additionally, we introduce an oracle, with access to which one can learn the number of components of $G$ in asymptotically fewer queries than learning the full partition, thus answering another question posed by the same authors. Lastly, we introduce a more applicable version of this oracle, and prove asymptotically tight bounds of $\widetildeΘ(m)$ queries for both learning and verifying an $m$-edge hidden graph $G$ using it.