  Fuzzy constraints are a popular approach to handle preferences and
over-constrained problems in scenarios where one needs to be cautious, such as
in medical or space applications. We consider here fuzzy constraint problems
where some of the preferences may be missing. This models, for example,
settings where agents are distributed and have privacy issues, or where there
is an ongoing preference elicitation process. In this setting, we study how to
find a solution which is optimal irrespective of the missing preferences. In
the process of finding such a solution, we may elicit preferences from the user
if necessary. However, our goal is to ask the user as little as possible. We
define a combined solving and preference elicitation scheme with a large number
of different instantiations, each corresponding to a concrete algorithm which
we compare experimentally. We compute both the number of elicited preferences
and the "user effort", which may be larger, as it contains all the preference
values the user has to compute to be able to respond to the elicitation
requests. While the number of elicited preferences is important when the
concern is to communicate as little information as possible, the user effort
measures also the hidden work the user has to do to be able to communicate the
elicited preferences. Our experimental results show that some of our algorithms
are very good at finding a necessarily optimal solution while asking the user
for only a very small fraction of the missing preferences. The user effort is
also very small for the best algorithms. Finally, we test these algorithms on
hard constraint problems with possibly missing constraints, where the aim is to
find feasible solutions irrespective of the missing constraints.
