A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

الأصل · EN

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.