Arithmetic Progression Hypergraphs: Examining the Second Moment Method

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 2coloring 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. ∗Harvard University, School of Engineering and Applied Sciences. Supported in part by NSF grants CCF-1320231, CNS1228598, IIS-0964473, and CCF-0915922. Part of this work was done while visiting Microsoft Research, New England.