Optimal Graph Reconstruction by Counting Connected Components in Induced Subgraphs

TL;DR

This paper introduces a new query model for graph reconstruction based on counting connected components in induced subgraphs, establishing optimal bounds for adaptive and non-adaptive queries.

Contribution

It proposes a novel query model for graph reconstruction and determines tight bounds for the number of queries needed in adaptive and non-adaptive settings.

Findings

Theta(m log n / log m) queries are sufficient and necessary for adaptive reconstruction.

Omega(n^2) non-adaptive queries are required even for sparse graphs.

An efficient two-round adaptive algorithm with O(m log n + n log^2 n) queries is provided.

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 $O(m\log n + n\log^2 n)$ query algorithm using only two rounds of adaptivity.