We study the problem of approximability of nondense instances of MAX-CUT. This complements the stream of results on existence of PTASs for dense or subdense instances of that problem (cf.[AKK95],[F96],[FK96],[GGR96],[FK00],[K01],[AFKK02],[FK02],[B05]). The paper aims at clarifying the inherent limits for existence of PTASs for instances of MAX-CUT which are parameterized by their densities. The algorithmic connections with Szemerédi’s Regularity Lemma discovered in [FK96], raise also questions on their algorithmic nondense analogs. There were also recent breakthrough extensions of Szemerédi’s Theorem to nondense settings dealing with prime numbers (cf.[GT06]). Back to the approximability of MAX-CUT on instances of arbitrary density. It is known that the (strongly) sparse instances of MAX-CUT with Θ (n) edges can be approximated with better factor ([FKL02]) than 1.1383 of Goemans-Williamson approximation algorithm ([GW95]). On the other hand, it is known that the

∗ LRI, CNRS, Université de Paris-Sud. Research partially supported by the IST grant 14036 (RAND-APX), and by the PROCOPE project. Email: lalo@ lri. fr.† Dept. of Computer Science, University of Bonn. Research partially done while visiting CSAIL, MIT and Microsoft Research, Redmond. Research partially supported by DFG grant, PROCOPE project, and by IST grant 14036 (RAND-APX). Email: marek@ cs. uni-bonn. de