A matter of preference

From Incomplete Preferences to Ranking via Optimization
P. Chebotarev, E. Shamis

This fascinating problem deals with how to determine a full ranking of a finite list of candidates given only partial preference data. For instance, if I prefer cola to coffee and green tea to black tea, and you prefer coffee to black tea, the following are compatible rankings for these preference data:

1. cola
2. coffee
3. green tea
4. black tea

and

1. cola
2. coffee
3. black tea
4. green tea

In this simple example there are two possible rankings; but in real-world situations, there are often none. How can one use a ranking to best approximate sets of incomplete prefence data, given that there will probably be no single ranking consistent with all preferences? I (and many others) would really like to know.