Planted matching problems on random hypergraphs

TL;DR

This paper investigates the detectability of hidden matchings in weighted random hypergraphs, revealing a phase transition in recoverability that depends on hypergraph structure and hyperedge composition.

Contribution

It introduces a phase transition framework for matching recovery in hypergraphs, highlighting differences between hypergraph orders and mixed structures.

Findings

First order transition for $k>2$ hypergraphs in recovery

Continuous transition for $k=2$ hypergraphs

Transition type changes with hyperedge mixture

Abstract

We consider the problem of inferring a matching hidden in a weighted random $k$-hypergraph. We assume that the hyperedges' weights are random and distributed according to two different densities conditioning on the fact that they belong to the hidden matching, or not. We show that, for $k>2$ and in the large graph size limit, an algorithmic first order transition in the signal strength separates a regime in which a complete recovery of the hidden matching is feasible from a regime in which partial recovery is possible. This is in contrast to the $k=2$ case where the transition is known to be continuous. Finally, we consider the case of graphs presenting a mixture of edges and $3$-hyperedges, interpolating between the $k=2$ and the $k=3$ cases, and we study how the transition changes from continuous to first order by tuning the relative amount of edges and hyperedges.