Finding Approximate Solutions to NP-Hard Problems by Neural Networks is Hard

Finding approximate solutions to hard combinatorial optimization problems by neu-ral 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 NP6 =co-NP. This paper shows that even nding approximate solutions with guaranteed performance to some NP-hard problems by a polynomial size network is also impossible unless NP=co-NP.