Peeling Algorithm on Random Hypergraphs with Superlinear Number of Hyperedges

TL;DR

This paper analyzes the peeling algorithm on random hypergraphs with superlinear hyperedges, deriving a tight threshold using a novel method since traditional techniques are inadequate.

Contribution

It introduces the evolution of the moment generating function method to analyze the peeling algorithm on hypergraphs with superlinear hyperedges, overcoming previous limitations.

Findings

Derived the tight threshold for the peeling algorithm on superlinear hypergraphs

Proposed a new analysis method called evolution of the moment generating function

Extended understanding of peeling algorithm dynamics beyond linear hyperedge regimes

Abstract

When we try to solve a system of linear equations, we can consider a simple iterative algorithm in which an equation including only one variable is chosen at each step, and the variable is fixed to the value satisfying the equation. The dynamics of this algorithm is captured by the peeling algorithm. Analyses of the peeling algorithm on random hypergraphs are required for many problems, e.g., the decoding threshold of low-density parity check codes, the inverting threshold of Goldreich's pseudorandom generator, the load threshold of cuckoo hashing, etc. In this work, we deal with random hypergraphs including superlinear number of hyperedges, and derive the tight threshold for the succeeding of the peeling algorithm. For the analysis, Wormald's method of differential equations, which is commonly used for analyses of the peeling algorithm on random hypergraph with linear number of hyperedges, cannot be used due to the superlinear number of hyperedges. A new method called the evolution of the moment generating function is proposed in this work.