Finding approximate solutions to hard combinatorial optimization problems by neural networks is a very attractive prospect. Many empirical studies have been done in the area. However, recent research about a neural network model indicates that for any NP-hard problem the existance of a polynomial size network that solves it implies that NP=co-NP, which is contrary to the well-known conjecture that<F NaN> NP6=co-NP. This paper shows that even finding approximate solutions with guaranteed performance to some NP-hard problems by a polynomial size network is also impossible unless NP=co-NP. Keywords --- Neural Networks, Combinatorial Optimization, Computational Complexity. 1 Introduction The interest in mapping combinatorial optimization problems onto neural networks has been growing rapidly since Hopfield and Tank first used them to solve TSP . It has been demonstrated that neural network optimization algorithms can give good near optimal solutions to rather large NP-hard problems [..