Derandomized concentration bounds for polynomials, and hypergraph maximal independent set

TL;DR

This paper improves randomized parallel algorithms for finding maximal independent sets in hypergraphs, introduces derandomized concentration bounds for polynomials, and provides the first deterministic sub-polynomial time algorithm for hypergraphs of rank greater than 3.

Contribution

It presents a faster randomized algorithm for hypergraph MIS and introduces a derandomization technique for concentration bounds, leading to the first deterministic sub-polynomial algorithm for high-rank hypergraphs.

Findings

Reduced runtime of randomized hypergraph MIS algorithm to $(\,\log n)^{2^r}$

Developed a derandomization method for concentration bounds of low-degree polynomials

First deterministic sub-polynomial time algorithm for hypergraphs with rank > 3

Abstract

A parallel algorithm for maximal independent set (MIS) in hypergraphs has been a long-standing algorithmic challenge, dating back nearly 30 years to a survey of Karp & Ramachandran (1990). The best randomized parallel algorithm for hypergraphs of fixed rank $r$ was developed by Beame & Luby (1990) and Kelsen (1992), running in time roughly $(\log n)^{r!}$. We improve the randomized algorithm of Kelsen, reducing the runtime to roughly $(\log n)^{2^r}$ and simplifying the analysis through the use of more-modern concentration inequalities. We also give a method for derandomizing concentration bounds for low-degree polynomials, which are the key technical tool used to analyze that algorithm. This leads to a deterministic PRAM algorithm also running in $(\log n)^{2^{r+3}}$ time and $\text{poly}(m,n)$ processors. This is the first deterministic algorithm with sub-polynomial runtime for hypergraphs of rank $r > 3$. Our analysis can also apply when $r$ is slowly growing; using this in conjunction with a strategy of Bercea et al. (2015) gives a deterministic MIS algorithm running in time $\exp(O( \frac{\log (mn)}{\log \log (mn)}))$.