Query Complexity of Hypergraph Connectivity and Learnability using CUT Oracles

2026-07-01

AI Summary

This work addresses the structural non-identifiability of hypergraphs under cut (CUT) queries, which precludes exact edge-set reconstruction or connectivity determination in O(n) queries as achievable for ordinary graphs. To overcome this barrier, the authors propose a zero-error randomized algorithm that leverages “independent families” in conjunction with techniques from weighted graph connectivity, Möbius inversion, and symmetric submodular function minimization to efficiently uncover hypergraph structure. Key contributions include establishing a connection between hypergraph learnability and edge parity, obtaining k-connectivity certificates for r-bounded even-parity hypergraphs using Õ_r(kn) queries, breaking the general quadratic lower bound by reducing the query complexity for linear hypergraphs to Õ(kn^{1.5}), and achieving expected O(n)-query zero-error identification of connected components.

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

Problem

• hypergraph connectivity
• CUT oracle
• query complexity
• learnability
• identifiability barrier

Innovation

• CUT oracle
• hypergraph connectivity
• independent families
• Möbius transform
• query complexity