Derandomized Concentration Bounds for Polynomials, and Hypergraph Maximal Independent Set

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 and Ramachandran (1990). The best randomized parallel algorithm for hypergraphs of fixed rank r was developed by Beame and 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)(2r) 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)(2r+3) time and 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(log(mn)/log log(mn))).