Abstract
One of the earliest and best-known application of the probabilistic method is the proof of existence of a 2 logn-Ramsey graph, i.e., a graph with n nodes that contains no clique or independent set of size 2 logn. The explicit construction of such a graph is a major open problem. We show that a reasonable hardness assumption implies that in polynomial time one can construct a list containing polylog(n) graphs such that most of them are 2 logn-Ramsey.
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References
Alon, N., Goldreich, O., Håstad, J., Peralta, R.: Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms 3(3), 289–304 (1992)
Barak, B., Rao, A., Shaltiel, R., Wigderson, A.: 2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction. In: Kleinberg, J.M. (ed.) STOC, pp. 671–680. ACM (2006)
Fortnow, L.: Full derandomization. Computational Complexity blog (July 31, 2006)
Frankl, P., Wilson, R.M.: Intersection theorems with geometric consequences. Combinatorica 1(4), 357–368 (1981)
Gasarch, W.I., Haeupler, B.: Lower bounds on van der Waerden numbers: Randomized- and deterministic-constructive. Electr. J. Comb. 18(1) (2011)
Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (STOC 1997), pp. 220–229. Association for Computing Machinery, New York (1997)
Klivans, A., van Melkebeek, D.: Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput. 31(5), 1501–1526 (2002)
Miltersen, P.B.: Derandomizing complexity classes. In: Pardalos, P., Reif, J., Rolim, J. (eds.) Handbook on Randomized Computing, Volume II. Kluwer Academic Publishers (2001)
Moser, R.A.: A constructive proof of the Lovász local lemma. In: Mitzenmacher, M. (ed.) STOC, pp. 343–350. ACM (2009)
Moore, C., Russell, A.: Optimal epsilon-biased sets with just a little randomness. CoRR, abs/1205.6218 (2012)
Moser, R.A., Tardos, G.: A constructive proof of the general Lovász local lemma. J. ACM 57(2) (2010)
Naor, M.: Constructing Ramsey graphs from small probability spaces. Technical report, IBM Research Report RJ 8810 (70940) (1992)
Naor, J., Naor, M.: Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing 22(4), 838–856 (1993)
Nisan, N., Wigderson, A.: Hardness vs. randomness. Journal of Computer and System Sciences 49, 149–167 (1994)
Santhanam, R.: The complexity of explicit constructions. Theory Comput. Syst. 51(3), 297–312 (2012)
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Zimand, M. (2013). On Efficient Constructions of Short Lists Containing Mostly Ramsey Graphs. In: Chan, TH.H., Lau, L.C., Trevisan, L. (eds) Theory and Applications of Models of Computation. TAMC 2013. Lecture Notes in Computer Science, vol 7876. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38236-9_19
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DOI: https://doi.org/10.1007/978-3-642-38236-9_19
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