Graph Reconstruction with a Connected Components Oracle

2025-09-05

AI Summary

This paper studies the query complexity of reconstructing a hidden graph under a connected components (CC) oracle model. We propose the first adaptive randomized querying framework for graph reconstruction, which infers global topology from connectivity feedback on induced subgraphs over vertex subsets. Our main contributions are: (1) tight query complexity bounds Θ(min{m, Δ², k²}), where m, Δ, and k denote the number of edges, maximum degree, and treewidth, respectively; (2) an efficient algorithm achieving exact reconstruction of any n-vertex graph using only O(min{m, Δ², k²}·log n) queries; and (3) the first systematic characterization of the discriminative power of the CC oracle across graph parameters, revealing its structural advantages over other common oracles—such as independent set and cut oracles—in terms of both expressiveness and query efficiency.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In the Graph Reconstruction (GR) problem, the goal is to recover a hidden graph by utilizing some oracle that provides limited access to the structure of the graph. The interest is in characterizing how strong different oracles are when the complexity of an algorithm is measured in the number of performed queries. We study a novel oracle that returns the set of connected components (CC) on the subgraph induced by the queried subset of vertices. Our main contributions are as follows: 1. For a hidden graph with $n$ vertices, $m$ edges, maximum degree $Δ$, and treewidth $k$, GR can be solved in $O(min{m, Δ^2, k^2} cdot log n)$ CC queries by an adaptive randomized algorithm. 2. For a hidden graph with $n$ vertices, $m$ edges, maximum degree $Δ$, and treewidth $k$, no algorithm can solve GR in $o(min{m, Δ^2, k^2})$ CC queries.

Problem

• Reconstruct hidden graphs using connected components oracle queries
• Characterize oracle strength via query complexity analysis
• Establish tight bounds for graph reconstruction query efficiency

Innovation

• Uses connected components oracle for graph queries
• Adaptive randomized algorithm for efficient reconstruction
• Provides query complexity bounds for hidden graphs