A new multilayered PCP and the hardness of hypergraph vertex cover

Authors:
Irit Dinur;Venkatesan Guruswami;Subhash Khot;Oded Regev
Affiliations:
NEC Research Institute, Princeton, NJ;University of Washington, Seattle, WA;Princeton University, Princeton, NJ;Institute for Advanced Study, Princeton, NJ
Venue:
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Year:
2003

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Abstract

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 ε0. 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.