Query Complexity of Hypergraph Connectivity and Learnability using CUT Oracles

Abstract

We investigate the power of CUT queries to reveal the structure of unknown hypergraphs. While simple graphs allow for optimal $O(n)$-query connectivity algorithms, hypergraphs face a fundamental identifiability barrier in that distinct hypergraphs can share identical cut-profiles, making exact edge learning impossible in general, a primitive crucial in the graph connectivity algorithms. We first present a zero-error randomized algorithm that identifies the connected components of any weighted hypergraph using $O(n)$ expected queries, matching the $\hat{I}\text{c◯}(n)$ lower bound. This approach bypasses the reconstruction barrier by introducing the notion of ``independent families'' -- vertex subpartitions that do not share hyperedges -- and iteratively coarsening them using auxiliary weighted graph connectivity techniques [Liao-Chakrabarty, 2024]. Second, we demonstrate that the impossibility of exact learning depends on hyperedge parity. For even-parity hypergraphs, we show that the structure is reconstructible using a Möbius transform on the CUT function to implement binary-search-style vertex identification. This yields deterministic algorithms for obtaining $k$-connectivity certificates for $r$-bounded even hypergraphs in ${\tilde{O}}_r(kn)$ queries. Finally, we bypass parity and rank constraints for linear hypergraphs, achieving a subquadratic $\tilde{O}(k{n}^{1.5})$ query complexity for $k$-connectivity. This significantly improves upon the general $\tilde{O}(n^2)$ bound derived via symmetric submodular function minimization.