Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1712.01781

Arithmetic Progression Hypergraphs: Examining the Second Moment Method

Michael Mitzenmacher
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Michael Mitzenmacher. 2017-12-05. Arithmetic Progression Hypergraphs: Examining the Second Moment Method. https://arxiv.org/abs/1712.01781

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗