3-hitting set on Bounded Degree Hypergraphs: Upper and Lower Bounds on the Kernel Size

We study upper and lower bounds on the kernel size for the 3-hitting set problem on hypergraphs of degree at most 3, denoted 33-hs. We first show that, unless P=NP, 3-3-hs on 3-uniform hypergraphs does not have a kernel of size at most 35k/19 > 1.8421k. We then give a 4k − k kernel for 3-3-hs that is computable in time O(k). This result improves the upper bound of 4k on the kernel size for 33-hs, given by Wahlström. We also show that the upper bound results on the kernel size for 3-3-hs can be generalized to the 3-hs problem on hypergraphs of bounded degree ∆, for any integer-constant ∆ > 3.