Computing Hitting Set Kernels By AC^0-Circuits

TL;DR

This paper demonstrates that hitting set kernels for hypergraphs can be computed in constant parallel time, refuting previous conjectures about the limits of parallelization, by introducing a new sunflower concept and advanced color coding techniques.

Contribution

It introduces a new hypergraph sunflower concept and shows how to compute hitting set kernels in constant parallel time, surpassing prior sequential and limited parallel algorithms.

Findings

Hitting set kernels can be computed in constant parallel time.

A new generalized sunflower notion aids in parallel kernelization.

Iterated color coding can be collapsed into a single step.

Abstract

Given a hypergraph $H = (V,E)$, what is the smallest subset $X \subseteq V$ such that $e \cap X \neq \emptyset$ holds for all $e \in E$? This problem, known as the hitting set problem, is a basic problem in parameterized complexity theory. There are well-known kernelization algorithms for it, which get a hypergraph $H$ and a number $k$ as input and output a hypergraph $H'$ such that (1) $H$ has a hitting set of size $k$ if, and only if, $H'$ has such a hitting set and (2) the size of $H'$ depends only on $k$ and on the maximum cardinality $d$ of edges in $H$. The algorithms run in polynomial time, but are highly sequential. Recently, it has been shown that one of them can be parallelized to a certain degree: one can compute hitting set kernels in parallel time $O(d)$ -- but it was conjectured that this is the best parallel algorithm possible. We refute this conjecture and show how hitting set kernels can be computed in constant parallel time. For our proof, we introduce a new, generalized notion of hypergraph sunflowers and show how iterated applications of the color coding technique can sometimes be collapsed into a single application.