Does anyone knows of any project management framework/methodology that has been developed specifically to cater to very small enterprises?
I know that Waterfall presents a serious overhead that cannot be tolerated by small enterprises.
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Have you had a look at Agile? It's a group of methods often used for small projects and / or rough requirements. Worth to read about (and then make your question more specific).
Notice that I don't believe there's such framework/methodology based on the company's size. We can have a huge company with tiny projects as well as a (relatively) small company with a (relatively) huge project.
I think you're misunderstanding the waterfall concept. Waterfall simply means that the project is sequenced, and that certain things must be done before other. In truth, all projects follow a waterfall process to some extent.
As for a small business, there's no specific methodology tailored to a type of 'business', methodologies are designed for types of projects. So what would be most applicable to you would be what suits your project best - software dev, then maybe Agile of Scrum; IT, maybe PRINCE2, const or fixed scope, then more waterfall/CPM.