Abstract
We consider the problem of reconstructing a graph G in two natural sampling models: 1) each sample corresponds to a random induced subgraph and 2) for a fixed adjacency matrix A_G for G, each sample corresponds to a random principal submatrix (i.e., a submatrix formed by deleting the same set of rows and columns) of A_G. We refer to these models as the "unordered" and "ordered" models respectively. The two models are motivated by work on the reconstruction conjecture in combinatorics and trace reconstruction in theoretical computer science. Despite the superficial similarities between the models, we show that the sample complexity of reconstruction can be exponentially different. Our main results are as follows:
 In the unordered model, we show that almost all graphs can be reconstructed with Θ(p^{2} log n) samples if each node is included in the random subgraph with any constant probability p; this is optimal. We show our upper bound extends to smaller values of p as well. In contrast, for arbitrary graphs, we show that exp(Ω(n)) samples are required for reconstruction even for 2regular graphs. One of the key technical steps in the first result is showing that, with high probability, any subgraph isomorphism in a random graph has at most O(log n) nonfixed points.
 In the ordered model, we show that any graph with constant arboricity or degeneracy (i.e., every induced subgraph has constant average degree) can be reconstructed with exp(Õ(n^{1/3})) samples and that arbitrary graphs can be reconstructed with exp(Õ(n^{1/2})) samples. The results about almost all graphs in the first model carry over to the second. One of the key technical steps in the first result is showing that reconstruction of low degeneracy graphs can be reduced to learning a small number of moments of sets of the form {ij: j < i,(i,j) ∈ E} and {ji: i < j,(i,j) ∈ E} where G = ([n],E) is the unknown graph.
BibTeX  Entry
@InProceedings{mcgregor_et_al:LIPIcs.ICALP.2022.96,
author = {McGregor, Andrew and Sengupta, Rik},
title = {{Graph Reconstruction from Random Subgraphs}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {96:196:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772358},
ISSN = {18688969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16437},
URN = {urn:nbn:de:0030drops164373},
doi = {10.4230/LIPIcs.ICALP.2022.96},
annote = {Keywords: graph reconstruction, sample complexity, deletion channel}
}
Keywords: 

graph reconstruction, sample complexity, deletion channel 
Collection: 

49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) 
Issue Date: 

2022 
Date of publication: 

28.06.2022 