The planted $k$-factor problem

TL;DR

This paper studies the problem of recovering a hidden k-factor in a weighted random graph, analyzing phase transitions between full and partial recovery based on signal-to-noise ratio.

Contribution

It introduces a new framework for understanding the planted k-factor problem and provides criteria for phase transition points in large graphs.

Findings

Identification of phase transition between full and partial recovery

Derivation of a criterion for the transition point

Connection to known problems like planted matching and TSP

Abstract

We consider the problem of recovering an unknown $k$-factor, hidden in a weighted random graph. For $k=1$ this is the planted matching problem, while the $k=2$ case is closely related to the planted travelling salesman problem. The inference problem is solved by exploiting the information arising from the use of two different distributions for the weights on the edges inside and outside the planted sub-graph. We argue that, in the large size limit, a phase transition can appear between a full and a partial recovery phase as function of the signal-to-noise ratio. We give a criterion for the location of the transition.