The Ancient Greeks recognized their standard of cooking as an important art form inspiring Archestratos, in 330 BC, to write the first cookbook. The area of ancient Greece now known as Southern Italy and Sicily has often been referred to as Magna Graecia (Greater Greece).  This is because many cities of Southern Italy and Sicily were founded by ancient Greek colonists beginning in the 7th century BC.  As a result, the area's cooking was forever influenced by the ancient Greeks. Later, Greek gastronomy influenced many of the Mediterranean cultures and when the Romans recognized this, they soon hired Greek cooks for their kitchens and Greek cultural tutors for their children.
So the Amalfi Coast, as is most of Southern Italy and Sicily, is Italian with much Greek influence, hence, the "Greco-Roman" term in the title. Together, these two powerhouse cultures form the backbone of the famous Mediterranean Diet.  Coming next year!