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Using the FGLSS-reduction to prove inapproximability results for minimum vertex cover in hypergraphs. (English) Zbl 1343.68094
Goldreich, Oded (ed.), Studies in complexity and cryptography. Miscellanea on the interplay between randomness and computation. In collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman. Berlin: Springer (ISBN 978-3-642-22669-4/pbk). Lecture Notes in Computer Science 6650, 88-97 (2011).
Summary: Using known results regarding PCP, we present simple proofs of the inapproximability of vertex cover for hypergraphs. Specifically, we show that
1 Approximating the size of the minimum vertex cover in $$O(1)$$-regular hypergraphs to within a factor of 1.99999 is NP-hard.
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 L. Trevisan [in: Proceedings of the thirty-third annual ACM symposium on theory of computing, STOC 2001. New York, NY: ACM Press. 453–461 (2001; Zbl 1323.90059)] and J. Holmerin [in: Proceedings of the thirty-fourth annual ACM symposium on theory of computing, STOC 2002. New York, NY: ACM Press. 544–552 (2002; Zbl 1192.68323)]), 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.
For the entire collection see [Zbl 1220.68005].

##### MSC:
 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 05C65 Hypergraphs 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 90C27 Combinatorial optimization
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