Is there any computer program that automatically analyzes harmony given the score?
By harmony, I of course mean, 'I', 'III, 'IV' etc.. and their connection.
If you use Sibelius, it has a plug-in called "Add Chord Symbols" which will analyze the music and then notate chords according to the chord name. But it will not assign functional harmony analysis to the chords. For example, the plug-in will analyze all the notes in all parts in your arrangement and tell you that the chords are C, F, G7, C, and you will have to deduce that because you are in the key of C major, the chords are I, IV, V7, I.
If you want to automate the deduction process, after you run "Add Chord Symbols", you can run a second plug-in called "Nashville Chord Numbers" that is not part of the standard installation (you can download and install it yourself). It will analyze harmonic function for you, given some limits. The Nashville Number System is a simple form of harmonic analysis. Here is a blog post about this plug-in.
If you use Sibelius, check out the huge list of plug-ins created by developer Bob Zawalich (who created both the "Add Chord Symbols" and "Nashville Chord Numbers" plug-ins).