Arithmetic Progression Hypergraphs: Examining the Second Moment Method

TL;DR

This paper extends second moment method threshold arguments to arithmetic progression hypergraphs, demonstrating that double hashing techniques can be analyzed in this quasi-random hypergraph model for problems like 3-NAE-SAT and hypergraph 2-coloring.

Contribution

It introduces a novel AP hypergraph model and shows that second moment methods apply to double hashing variants of classical problems, broadening their analytical scope.

Findings

Second moment arguments extend to AP hypergraphs for 3-NAE-SAT.

Second moment arguments extend to AP hypergraphs for 2-coloring of 3-regular hypergraphs.

Introduces a unified quasi-random hypergraph model based on arithmetic progressions.

Abstract

In many data structure settings, it has been shown that using "double hashing" in place of standard hashing, by which we mean choosing multiple hash values according to an arithmetic progression instead of choosing each hash value independently, has asymptotically negligible difference in performance. We attempt to extend these ideas beyond data structure settings by considering how threshold arguments based on second moment methods can be generalized to "arithmetic progression" versions of problems. With this motivation, we define a novel "quasi-random" hypergraph model, random arithmetic progression (AP) hypergraphs, which is based on edges that form arithmetic progressions and unifies many previous problems. Our main result is to show that second moment arguments for 3-NAE-SAT and 2-coloring of 3-regular hypergraphs extend to the double hashing setting. We leave several open problems related to these quasi-random hypergraphs and the thresholds of associated problem variations.