The planted k-factor problem

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 traveling 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.

Planted matching problems on random hypergraphs

“…The inference problem we consider is given on an ensemble of (weighted) random hypergraphs which generalizes the ensemble of weighted graphs discussed in Refs [13,18]. This ensemble, which we denote H N k,c [ p, p], uses as input the coordination k of the hyperedges, an integer N ∈ N, two absolutely continuous probability densities p and p, and a real number c ∈ R + .…”

“…Moreover, the transition is found to be continuous and, for a specific choice of p and p, proven to be of infinite order. Interestingly, it has been shown, at the level of rigor of theoretical physics, that the phenomenology extends to the so-called planted kfactor problem [17,18], in which the hidden structure is a k factor of the graph, that is, a k-regular subgraph including all the nodes.…”