Abstract
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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1129-1146 |
| Number of pages | 18 |
| Journal | SIAM Journal on Computing |
| Volume | 34 |
| Issue number | 5 |
| DOIs | |
| State | Published - 2005 |
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Keywords
- Hardness of approximation
- Hypergraph vertex cover
- Long-code
- Multilayered outer verifier
- Probabilistically checkable proof
ASJC Scopus subject areas
- Theoretical Computer Science
- Computational Theory and Mathematics
- Applied Mathematics
Cite this
A new multilayered PCP and the hardness of hypergraph vertex cover. / Dinur, Irit; Guruswami, Venkatesan; Khot, Subhash; Regev, Oded.
In: SIAM Journal on Computing, Vol. 34, No. 5, 2005, p. 1129-1146.Research output: Contribution to journal › Article
}
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
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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UR - http://www.scopus.com/inward/citedby.url?scp=27144509145&partnerID=8YFLogxK
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 -