TY - JOUR

T1 - A new multilayered PCP and the hardness of hypergraph vertex cover

AU - Dinur, Irit

AU - Guruswami, Venkatesan

AU - Khot, Subhash

AU - Regev, Oded

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2005

Y1 - 2005

N2 - 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 probabilistically checkable proof (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 superconstant values of k up to (log N) 1/c, where c > 1 is a fixed constant and N is the number of hyperedges.

AB - 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 probabilistically checkable proof (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 superconstant values of k up to (log N) 1/c, where c > 1 is a fixed constant and N is the number of hyperedges.

KW - Hardness of approximation

KW - Hypergraph vertex cover

KW - Long-code

KW - Multilayered outer verifier

KW - Probabilistically checkable proof

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U2 - 10.1137/S0097539704443057

DO - 10.1137/S0097539704443057

M3 - Article

AN - SCOPUS:27144509145

VL - 34

SP - 1129

EP - 1146

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 5

ER -