Lower bounds for graph reconstruction with maximal independent set queries

TL;DR

The paper investigates the query complexity of reconstructing a hidden graph using maximal independent set queries, distinguishing adaptive vs non-adaptive and randomised vs deterministic paradigms. It develops a cover-free family framework to translate non-adaptive deterministic lower bounds into graph-reconstruction limits and employs information-counting arguments to derive lower bounds for randomised settings. The main results are $Ω(Δ^2 \log(n/Δ)/\log Δ)$ for randomised adaptive, $Ω(Δ^2 \log(n/Δ))$ for randomised non-adaptive, and $Ω(Δ^3 \log n / \log Δ)$ for deterministic non-adaptive, improving previous bounds and answering open questions. These findings clarify the gap between adaptive and non-adaptive strategies and highlight the role of combinatorial structures like cover-free families in determining query complexity, while leaving several tightness and algorithm-design questions open.

Abstract

We investigate the number of maximal independent set queries required to reconstruct the edges of a hidden graph. We show that randomised adaptive algorithms need at least $Ω(Δ^2 \log(n / Δ) / \log Δ)$ queries to reconstruct $n$-vertex graphs of maximum degree $Δ$ with success probability at least $1/2$, and we further improve this lower bound to $Ω(Δ^2 \log(n / Δ))$ for randomised non-adaptive algorithms. We also prove that deterministic non-adaptive algorithms require at least $Ω(Δ^3 \log n / \log Δ)$ queries. This improves bounds of Konrad, O'Sullivan, and Traistaru, and answers one of their questions. The proof of the lower bound for deterministic non-adaptive algorithms relies on a connection to cover-free families, for which we also improve known bounds.

Theorems & Definitions (21)

• Theorem 1.1: Konrad, O'Sullivan, and Traistaru
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Konrad, O'Sullivan, and Traistaru
• Theorem 1.5
• Lemma 1
• Theorem 1.6: Füredi
• Theorem 1.7
• Lemma 2
• proof