Reconstruction of graph colourings

TL;DR

This paper investigates the conditions under which random graph colourings can be reconstructed from their subgraph decks, providing precise thresholds for grids and random graphs, and offering efficient algorithms for reconstruction.

Contribution

It offers new almost linear algorithms for reconstructing colourings of grids and random graphs, and establishes sharp thresholds for when reconstruction is possible or impossible.

Findings

Reconstruction of grid colourings is possible for k ≥ (d log_r n)^{1/d} + 1/d + ε.

Reconstruction of random graph colourings is possible for k ≥ 2 log_2 n + 8.

Reconstruction is not possible below these thresholds with high probability.

Abstract

A $k$-deck of a (coloured) graph is a multiset of its induced $k$-vertex subgraphs. Given a graph $G$, when is it possible to reconstruct with high probability a uniformly random colouring of its vertices in $r$ colours from its $k$-deck? In this paper, we study this question for grids and random graphs. Reconstruction of random colourings of $d$-dimensional $n$-grids from the deck of their $k$-subgrids is one of the most studied colour reconstruction questions. The 1-dimensional case is motivated by the problem of reconstructing DNA sequences from their `shotgunned' stretches. It was comprehensively studied and the above reconstruction question was completely answered in the '90s. In this paper, we get a very precise answer for higher $d$. For every $d\geq 2$ and every $r\geq 2$, we present an almost linear algorithm that reconstructs with high probability a random $r$-colouring of vertices of a $d$-dimensional $n$-grid from the deck of all its $k$-subgrids for every $k\geq(d\log_r n)^{1/d}+1/d+\varepsilon$ and prove that the random $r$-colouring is not reconstructible with high probability if $k\leq (d\log_r n)^{1/d}-\varepsilon$. This answers the question of Narayanan and Yap (that was asked for $d\geq 3$) on "two-point concentration" of the minimum $k$ so that $k$-subgrids determine the entire colouring. Next, we prove that with high probability a uniformly random $r$-colouring of vertices of a uniformly random graph $G(n,1/2)$ is reconstructible from its full $k$-deck if $k\geq 2\log_2 n+8$ and is not reconstructible with high probability if $k\leq\sqrt{2\log_2 n}$. We further show that the colour reconstruction algorithm for random graphs can be modified and used for graph reconstruction: we prove that with high probability $G(n,1/2)$ is reconstructible from its full $k$-deck if $k\geq 2\log_2 n+11$ while it is not reconstructible with high probability if $k\leq 2\sqrt{\log_2 n}$.