Shotgun assembly of labeled graphs

TL;DR

This paper studies the problem of reconstructing labeled graphs from local neighborhoods, covering theoretical conditions for accurate recovery across various models including DNA, neural networks, and puzzles.

Contribution

It introduces new conditions for graph reconstruction and explores their applicability to different models, including random graphs and puzzles.

Findings

Derived necessary and sufficient conditions for graph recovery.

Analyzed reconstruction in models like lattice labelings, Erdos-Renyi graphs, and jigsaw puzzles.

Presented open problems and conjectures for future research.

Abstract

We consider the problem of reconstructing graphs or labeled graphs from neighborhoods of a given radius r. Special instances of this problem include the well known: DNA shotgun assembly; the lesser-known: neural network reconstruction; and a new problem: assembling random jigsaw puzzles. We provide some necessary and some sufficient conditions for correct recovery both in combinatorial terms and for some generative models including random labelings of lattices, Erdos-Renyi random graphs, and the random jigsaw puzzle model. Many open problems and conjectures are provided.