Query Complexity of Hypergraph Connectivity and Learnability using CUT Oracles

Deeparnab Chakrabarty; Hang Liao

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arXiv · 2607.01216

Query Complexity of Hypergraph Connectivity and Learnability using CUT Oracles

Deeparnab Chakrabarty·Hang Liao

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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 $\Omega(n)$ 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 M\"obius 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 $\tilde{O}(kn^{1.5})$ query complexity for $k$-connectivity. This significantly improves upon the general $\tilde{O}(n^2)$ bound derived via symmetric submodular function minimization.

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BibTeX RIS

Deeparnab Chakrabarty, Hang Liao. 2026-07-01. Query Complexity of Hypergraph Connectivity and Learnability using CUT Oracles. https://arxiv.org/abs/2607.01216

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