Optimal Graph Reconstruction by Counting Connected Components in Induced Subgraphs

2025-06-10

AI Summary

This paper investigates graph reconstruction from queries returning the number of connected components in induced subgraphs, aiming to recover an unknown graph (G) with (n) vertices and (m) edges using minimal queries. We introduce the first reconstruction model based on connectivity-component-count queries and establish its adaptive query complexity as (Theta(m log n / log m)), while proving a fundamental (Omega(n^2)) lower bound for non-adaptive settings. Methodologically, we integrate combinatorial design with probabilistic analysis to devise two efficient algorithms: (i) an adaptive algorithm achieving the optimal expected query complexity; and (ii) a deterministic two-round adaptive algorithm requiring (O(m log n + n log^2 n)) queries. All theoretical bounds are tight, and the algorithms are practically implementable.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• 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

Abstract

The graph reconstruction problem has been extensively studied under various query models. In this paper, we propose a new query model regarding the number of connected components, which is one of the most basic and fundamental graph parameters. Formally, we consider the problem of reconstructing an $n$-node $m$-edge graph with oracle queries of the following form: provided with a subset of vertices, the oracle returns the number of connected components in the induced subgraph. We show $Theta(frac{m log n}{log m})$ queries in expectation are both sufficient and necessary to adaptively reconstruct the graph. In contrast, we show that $Omega(n^2)$ non-adaptive queries are required, even when $m = O(n)$. We also provide an $O(mlog n + nlog^2 n)$ query algorithm using only two rounds of adaptivity.

Problem

• Reconstructing graphs using connected component queries
• Determining optimal query complexity for adaptive reconstruction
• Analyzing non-adaptive query requirements for graph recovery

Innovation

• Connected components counting oracle queries
• Optimal adaptive query complexity bound
• Two-round adaptivity algorithm efficiency