Given a k-uniform hyper-graph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within factor (k−1−ϵ) for any k≥3 and any ϵ>0. The result is essentially tight as this problem can be easily approximated within factor k. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets.
@article{arxiv.cs/0304026,
title = {A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover},
author = {Irit Dinur and Venkatesan Guruswami and Subhash Khot and Oded Regev},
journal= {arXiv preprint arXiv:cs/0304026},
year = {2007}
}