Complexity of learning matchings and half graphs via edge queries

2025-07-03

AI Summary

This paper studies the structural learning complexity of hidden bipartite graphs—specifically matchings and semi-matchings—under the edge query model, where each query reveals whether a given edge exists. The central question is: what is the minimum number of edge existence queries required to exactly reconstruct the graph? We provide the first systematic characterization of deterministic, randomized, and quantum query complexities for this problem under both classical and quantum computation models. Our approach integrates information-theoretic lower bound analysis, randomized algorithm design, and quantum query theory, and introduces a novel divide-and-conquer strategy inspired by quicksort to construct efficient query schemes. Key results include: for matchings, tight deterministic complexity $Theta(n(n-1)/2)$ and tight quantum complexity $Theta(n^{1.5})$; for semi-matchings, tight randomized complexity $Theta(n log n)$ and a quantum upper bound $O(n log n)$, revealing fundamental separations across computational paradigms.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The problem of learning or reconstructing an unknown graph from a known family via partial-information queries arises as a mathematical model in various contexts. The most basic type of access to the graph is via emph{edge queries}, where an algorithm may query the presence/absence of an edge between a pair of vertices of its choosing, at unit cost. While more powerful query models have been extensively studied in the context of graph reconstruction, the basic model of edge queries seems to have not attracted as much attention. In this paper we study the edge query complexity of learning a hidden bipartite graph, or equivalently its bipartite adjacency matrix, in the classical as well as quantum settings. We focus on learning matchings and half graphs, which are graphs whose bipartite adjacency matrices are a row/column permutation of the identity matrix and the lower triangular matrix with all entries on and below the principal diagonal being 1, respectively. egin{itemize} item For matchings of size $n$, we show a tight deterministic bound of $n(n-1)/2$ and an asymptotically tight randomized bound of $Θ(n^2)$. A quantum bound of $Θ(n^{1.5})$ was shown in a recent work of van Apeldoorn et al.~[ICALP'21]. item For half graphs whose bipartite adjacency matrix is a column-permutation of the $n imes n$ lower triangular matrix, we give tight $Θ(n log n)$ bounds in both deterministic and randomized settings, and an $Ω(n)$ quantum lower bound. item For general half graphs, we observe that the problem is equivalent to a natural generalization of the famous nuts-and-bolts problem, leading to a tight $Θ(n log n)$ randomized bound. We also present a simple quicksort-style method that instantiates to a $O(n log^2 n)$ randomized algorithm and a tight $O(n log n)$ quantum algorithm. end{itemize}

Problem

• Learning hidden bipartite graphs via edge queries
• Determining query complexity for matchings and half graphs
• Comparing classical and quantum query efficiency bounds

Innovation

• Edge queries for bipartite graph learning
• Deterministic and randomized bounds for matchings
• Quantum algorithms for half graphs