Abstract
It was previously known that Max Clique cannot be approximated in polynomial time within n1-ε, for any constant ε > 0, unless NP = ZPP. In this paper, we extend the reductions used to prove this result and combine the extended reductions with a recent result of Samorodnitsky and Trevisan to show that clique cannot be approximated within \( n^{1 - O} \left( {1/\sqrt {\log \log n} } \right) \) unless NP ⊆ ZPTIMENP ⊆ ZPTIME(2O(log n(log log n)3/2)).
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References
S. Arora, C. Lund, R. Motwani, M. Sudhan, and M. Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501–555, May 1998.
S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70–122, Jan. 1998.
M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCPs and non-approximability—towards tight results. SIAM J. Comput., 27(3):804–915, June 1998.
M. Bellare and M. Sudan. Improved non-approximability results. In Proc. Twenty-sixth Ann. ACM Symp. on Theory of Comp., pages 184–193, Montréal, Québec, May 1994. ACM Press.
R. Boppana and M. M. Halldórsson. Approximating maximum independent sets by excluding subgraphs. Bit, 32(2):180–196, June1992.
U. Feige. Randomized graph products, chromatic numbers, and the Lovasz ϑ-function. In Proc. Twenty-seventh Ann. ACM Symp. on Theory of Comp., pages 635–640, Las Vegas, Nevada, May 1995. ACM Press.
U. Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634–652, July1998.
U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy. Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268–292, Mar. 1996.
J. Hästad. Clique is hard to approximate within n1-ε. In Proc. 37th Ann. IEEE Symp. on Foundations of Comput. Sci., pages 627–636, Burlington, Vermont, Oct. 1996. IEEE Computer Society.
R. M. Karp. Reducibility among combinatorial problems. In R.E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 85–103. Plenum Press, New York, 1972.
A. Samorodnitsky and L. Trevisan. Notes on a PCP characterization of NP with optimal amortized query complexity. Manuscript, May 1999.
D. Zuckerman. On unapproximable versions of NP-complete problems. SIAM J. Comput., 25(6):1293–1304, Dec. 1996.
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Engebretsen, L., Holmerin, J. (2000). Clique Is Hard to Approximate within n 1-o(1). In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds) Automata, Languages and Programming. ICALP 2000. Lecture Notes in Computer Science, vol 1853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45022-X_2
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DOI: https://doi.org/10.1007/3-540-45022-X_2
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