A new multilayered PCP and the hardness of hypergraph vertex cover

Given a k-uniform hypergraph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyperedge. 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 a factor of (k − 1 − ε) for arbitrary constants ε > 0 and k ≥ 3. The result is nearly 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. We also give a different proof that shows an inapproximability factor of ⌊k 2 ⌋ − ε. In addition to being simpler, this proof also works for super-constant values of k up to (log N)1/c where c > 1 is a fixed constant and N is the number of hyperedges.

• Type: Conference
• Year: 2003
• Where: STOC
• Authors: Irit Dinur, Venkatesan Guruswami, Subhash Khot, Oded Regev
• Keywords: Algorithms, K-uniform Hypergraph, Multilayered Pcp Construction, Smallest Subset, STOC 2003