3-Hitting set on bounded degree hypergraphs: Upper and lower bounds on the kernel size

Abstract

We study upper and lower bounds on the kernel size for the 3-hitting set problem on hypergraphs of degree at most 3, denoted 3-3-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 - k0.2692 kernel for 3-3-hs that is computable in time O(k1.2692). This result improves the upper bound of 4k on the kernel size for 3-3-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 Δ, for any integer-constant Δ > 3. © 2011 Springer-Verlag.