Abstract
Using known results regarding PCP, we present simple proofs of the inapproximability of vertex cover for hypergraphs. Specifically, we show that
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1
Approximating the size of the minimum vertex cover in O(1)-regular hypergraphs to within a factor of 1.99999 is NP-hard.
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2
Approximating the size of the minimum vertex cover in 4-regular hypergraphs to within a factor of 1.49999 is NP-hard.
Both results are inferior to known results (by Trevisan (2001) and Holmerin (2001)), but they are derived using much simpler proofs. Furthermore, these proofs demonstrate the applicability of the FGLSS-reduction in the context of reductions among combinatorial optimization problems.
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Goldreich, O. (2011). Using the FGLSS-Reduction to Prove Inapproximability Results for Minimum Vertex Cover in Hypergraphs. In: Goldreich, O. (eds) Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation. Lecture Notes in Computer Science, vol 6650. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22670-0_11
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DOI: https://doi.org/10.1007/978-3-642-22670-0_11
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