Determining relationships in data

Assumptions:

• The scores are generated by an unknown function
• Hundreds of variables in your data
• Each set of data has a score or rank
• The dataset is large, 10,000 - 1million entries of these sets of data/scores (the data set could be are large as required)

Is it possible to determine the contribution of each variable to the score? It's unknown what the weightings are.

Is it possible to determine the effect of one variable on another? It's known that a high or low value for variable n might change the contribution of variable k by some amount. But it isn't limited to linear relationships, variables a, b, and c could have their weightings increased by p, q, and r if the value for s and t are above/below some threshold.

I'm not sure its possible to do this, let alone do it in under exp/factorial time complexity... I hope the question is clear enough.