Computing Hitting Set Kernels by AC0-Circuits

Given a hypergraph H = (V,E), what is the smallest subset $X \subseteq V$ such that e ∩ X≠∅ holds for all e ∈ E? This problem, known as the hitting set problem, is a basic problem in combinatorial optimization and has been studied extensively in both classical and 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 $H^{\prime }$ has such a hitting set and (2) the size of $H^{\prime }$ depends only on k and on the maximum cardinality d of hyperedges in H. The algorithms run in polynomial time and can be parallelized to a certain degree: one can easily compute hitting set kernels in parallel time O(k) and not-so-easily in time O(d) – but it was conjectured that these are the best parallel algorithms 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.