Anthropology/Question upon Neanderthal extinction
What is the most acceptable or has the most evidence hypothesis for the Neanderthal extinction? Would it be the interbreeding between Humans and the Neanderthals?
Interbreeding is one of several proposed mechanisms for Neandertal extinction. Competitive exclusion by Homo sapiens is another. Extinction due to low population sizes and rapid climate change yet a third.
The difficulty with answering your question is that there were probably different causes with correspondingly different levels of importance in different times and different places. Extinction is a complex phenomenon.
Interbreeding might have been more important in Neandertal-Homo sapiens contacts in earlier times when the populations involved were closer to their last common ancestor and where there were relatively large populations involved, such as SW Asia or the Mediterranean watershed of Europe (higher terrestrial productivity = larger populations). Competitive exclusion and low-population/climate change might have been more important factors in other regions where resources were scarcer, such as Northern Europe.
The problem with this issue has much to do with HOW scientists are arguing about it. In popular media, we are arguing like attorneys -proposing ONE explanation and prosecuting/defending it. In actuality, most of us involved in research on this issue know that there are multiple possible explanations with situationally-variable probabilities. For my part, I think we have to go with what Glynn Isaac (the renowned African prehistorian) called the method of "multiple working hypotheses" -keep all plausible and potentially testable hypotheses on the table and work towards finding strong contrary evidence against them.
FWIW: The recent evidence of Neandertal genetic introgression ("Neandertal" DNA among living humans) suggests some interbreeding might have occurred, but to keep this evidence in perspective, the contributions of this Neandertal DNA are relatively modest and the residual DNA overwhelmingly that of ancestral African Homo sapiens.