Speaker: David Harris

Affiliation: University of Maryland College Park

Title: Improved parallel algorithms for hypergraph maximal independent set

Abstract:

Finding a maximal independent set in hypergraphs has been a long-standing algorithmic challenge. The best parallel algorithm for hypergraphs of rank $r$ was developed by Beame

and Luby (1990) and Kelsen (1992), running in time roughly $(\log n)^{r!}$. This is in RNC for fixed $r$, but is still quite expensive. We improve on the analysis of Kelsen to

show that (a slight variant) of this algorithm runs in time $(\log n)^{2^r}$. We derandomize this algorithm to achieve a deterministic algorithm running in time $(\log

n)^{2^{r+3}}$ using $m^{O(1)}$ processors.

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 algorithm running in time

$\exp(O(\log m/\log \log m))$. This is faster than the algorithm of Bercea et al, and in addition it is deterministic. In particular, this is sub-polynomial time for graphs with

$m \leq n^{o(\log \log n)}$ edges.